| L(s) = 1 | + 3·5-s + 4·7-s + 3·11-s − 9·13-s + 13·17-s + 4·23-s − 9·25-s + 9·29-s + 12·31-s + 12·35-s − 14·37-s + 11·41-s − 7·43-s + 8·47-s − 13·49-s + 22·53-s + 9·55-s + 9·59-s − 19·61-s − 27·65-s − 4·67-s + 15·71-s − 10·73-s + 12·77-s + 9·79-s + 25·83-s + 39·85-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 1.51·7-s + 0.904·11-s − 2.49·13-s + 3.15·17-s + 0.834·23-s − 9/5·25-s + 1.67·29-s + 2.15·31-s + 2.02·35-s − 2.30·37-s + 1.71·41-s − 1.06·43-s + 1.16·47-s − 1.85·49-s + 3.02·53-s + 1.21·55-s + 1.17·59-s − 2.43·61-s − 3.34·65-s − 0.488·67-s + 1.78·71-s − 1.17·73-s + 1.36·77-s + 1.01·79-s + 2.74·83-s + 4.23·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 139^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 139^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(33.29871681\) |
| \(L(\frac12)\) |
\(\approx\) |
\(33.29871681\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 139 | \( ( 1 - T )^{6} \) |
| good | 5 | \( 1 - 3 T + 18 T^{2} - 52 T^{3} + 191 T^{4} - 429 T^{5} + 1203 T^{6} - 429 p T^{7} + 191 p^{2} T^{8} - 52 p^{3} T^{9} + 18 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 7 | \( 1 - 4 T + 29 T^{2} - 106 T^{3} + 451 T^{4} - 1271 T^{5} + 4079 T^{6} - 1271 p T^{7} + 451 p^{2} T^{8} - 106 p^{3} T^{9} + 29 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 11 | \( 1 - 3 T + 34 T^{2} - 100 T^{3} + 555 T^{4} - 1479 T^{5} + 6643 T^{6} - 1479 p T^{7} + 555 p^{2} T^{8} - 100 p^{3} T^{9} + 34 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( 1 + 9 T + 77 T^{2} + 413 T^{3} + 2104 T^{4} + 8532 T^{5} + 33476 T^{6} + 8532 p T^{7} + 2104 p^{2} T^{8} + 413 p^{3} T^{9} + 77 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( 1 - 13 T + 111 T^{2} - 717 T^{3} + 4111 T^{4} - 20590 T^{5} + 92114 T^{6} - 20590 p T^{7} + 4111 p^{2} T^{8} - 717 p^{3} T^{9} + 111 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \) |
| 19 | \( 1 + 21 T^{2} + 24 T^{3} + 83 T^{4} + 1592 T^{5} - 5474 T^{6} + 1592 p T^{7} + 83 p^{2} T^{8} + 24 p^{3} T^{9} + 21 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 - 4 T + 77 T^{2} - 352 T^{3} + 3475 T^{4} - 13372 T^{5} + 99662 T^{6} - 13372 p T^{7} + 3475 p^{2} T^{8} - 352 p^{3} T^{9} + 77 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 - 9 T + 157 T^{2} - 1155 T^{3} + 11056 T^{4} - 62344 T^{5} + 426016 T^{6} - 62344 p T^{7} + 11056 p^{2} T^{8} - 1155 p^{3} T^{9} + 157 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( 1 - 12 T + 125 T^{2} - 814 T^{3} + 5043 T^{4} - 29473 T^{5} + 163571 T^{6} - 29473 p T^{7} + 5043 p^{2} T^{8} - 814 p^{3} T^{9} + 125 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( 1 + 14 T + 219 T^{2} + 53 p T^{3} + 18223 T^{4} + 121740 T^{5} + 850764 T^{6} + 121740 p T^{7} + 18223 p^{2} T^{8} + 53 p^{4} T^{9} + 219 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \) |
| 41 | \( 1 - 11 T + 34 T^{2} + 443 T^{3} - 2128 T^{4} - 16047 T^{5} + 257904 T^{6} - 16047 p T^{7} - 2128 p^{2} T^{8} + 443 p^{3} T^{9} + 34 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 + 7 T + 153 T^{2} + 15 p T^{3} + 10523 T^{4} + 32826 T^{5} + 515350 T^{6} + 32826 p T^{7} + 10523 p^{2} T^{8} + 15 p^{4} T^{9} + 153 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( 1 - 8 T + 77 T^{2} - 375 T^{3} - 561 T^{4} + 30412 T^{5} - 184122 T^{6} + 30412 p T^{7} - 561 p^{2} T^{8} - 375 p^{3} T^{9} + 77 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 - 22 T + 466 T^{2} - 6030 T^{3} + 72583 T^{4} - 648244 T^{5} + 5375100 T^{6} - 648244 p T^{7} + 72583 p^{2} T^{8} - 6030 p^{3} T^{9} + 466 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 - 9 T + 135 T^{2} - 1007 T^{3} + 9207 T^{4} - 102834 T^{5} + 683618 T^{6} - 102834 p T^{7} + 9207 p^{2} T^{8} - 1007 p^{3} T^{9} + 135 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 + 19 T + 405 T^{2} + 4995 T^{3} + 62543 T^{4} + 566022 T^{5} + 5083126 T^{6} + 566022 p T^{7} + 62543 p^{2} T^{8} + 4995 p^{3} T^{9} + 405 p^{4} T^{10} + 19 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 + 4 T + 209 T^{2} + 1438 T^{3} + 18311 T^{4} + 203511 T^{5} + 1180551 T^{6} + 203511 p T^{7} + 18311 p^{2} T^{8} + 1438 p^{3} T^{9} + 209 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 - 15 T + 391 T^{2} - 4279 T^{3} + 64656 T^{4} - 544860 T^{5} + 5953124 T^{6} - 544860 p T^{7} + 64656 p^{2} T^{8} - 4279 p^{3} T^{9} + 391 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( 1 + 10 T + 283 T^{2} + 2118 T^{3} + 39271 T^{4} + 232504 T^{5} + 3398362 T^{6} + 232504 p T^{7} + 39271 p^{2} T^{8} + 2118 p^{3} T^{9} + 283 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 - 9 T + 256 T^{2} - 2072 T^{3} + 31979 T^{4} - 226281 T^{5} + 35391 p T^{6} - 226281 p T^{7} + 31979 p^{2} T^{8} - 2072 p^{3} T^{9} + 256 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( 1 - 25 T + 390 T^{2} - 4236 T^{3} + 31309 T^{4} - 153217 T^{5} + 968117 T^{6} - 153217 p T^{7} + 31309 p^{2} T^{8} - 4236 p^{3} T^{9} + 390 p^{4} T^{10} - 25 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( 1 - 36 T + 11 p T^{2} - 18062 T^{3} + 278679 T^{4} - 3391629 T^{5} + 35507557 T^{6} - 3391629 p T^{7} + 278679 p^{2} T^{8} - 18062 p^{3} T^{9} + 11 p^{5} T^{10} - 36 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 + 47 T + 1225 T^{2} + 22503 T^{3} + 329207 T^{4} + 4040642 T^{5} + 42789262 T^{6} + 4040642 p T^{7} + 329207 p^{2} T^{8} + 22503 p^{3} T^{9} + 1225 p^{4} T^{10} + 47 p^{5} T^{11} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.48349358541471925224509820166, −4.02900719541539429256070280972, −3.79642106450382048630950795746, −3.75313670838482974026281096290, −3.75024686080485815753525205895, −3.49352097000120029149890541540, −3.48054390867179038749701993881, −3.09682117166956029164283551070, −3.00550191969056178409192389544, −2.92143967746670816134443036184, −2.81939515734328293881607923413, −2.76918079395572630855922201054, −2.35966625636834976316587664439, −2.11299037717579447696623386448, −2.03927730845738941127470603142, −2.01739354373388387485480407500, −1.97709677438191677396147086789, −1.67320831019983048169814943925, −1.65083469473984976726224469744, −1.19284031689961282736274503758, −1.08916922848664646103873236181, −0.924027356329474916233760308764, −0.73009918190155076879901604695, −0.49925448083348703020175015038, −0.43405793049354813334787140245,
0.43405793049354813334787140245, 0.49925448083348703020175015038, 0.73009918190155076879901604695, 0.924027356329474916233760308764, 1.08916922848664646103873236181, 1.19284031689961282736274503758, 1.65083469473984976726224469744, 1.67320831019983048169814943925, 1.97709677438191677396147086789, 2.01739354373388387485480407500, 2.03927730845738941127470603142, 2.11299037717579447696623386448, 2.35966625636834976316587664439, 2.76918079395572630855922201054, 2.81939515734328293881607923413, 2.92143967746670816134443036184, 3.00550191969056178409192389544, 3.09682117166956029164283551070, 3.48054390867179038749701993881, 3.49352097000120029149890541540, 3.75024686080485815753525205895, 3.75313670838482974026281096290, 3.79642106450382048630950795746, 4.02900719541539429256070280972, 4.48349358541471925224509820166
Plot not available for L-functions of degree greater than 10.
