| L(s) = 1 | + 2·4-s + 5-s − 2·11-s − 3·16-s + 6·19-s + 2·20-s + 2·25-s + 36·29-s − 12·41-s − 4·44-s + 23·49-s − 2·55-s + 20·59-s − 14·61-s − 8·64-s − 52·71-s + 12·76-s + 24·79-s − 3·80-s + 24·89-s + 6·95-s + 4·100-s − 20·101-s − 40·109-s + 72·116-s − 31·121-s + 125-s + ⋯ |
| L(s) = 1 | + 4-s + 0.447·5-s − 0.603·11-s − 3/4·16-s + 1.37·19-s + 0.447·20-s + 2/5·25-s + 6.68·29-s − 1.87·41-s − 0.603·44-s + 23/7·49-s − 0.269·55-s + 2.60·59-s − 1.79·61-s − 64-s − 6.17·71-s + 1.37·76-s + 2.70·79-s − 0.335·80-s + 2.54·89-s + 0.615·95-s + 2/5·100-s − 1.99·101-s − 3.83·109-s + 6.68·116-s − 2.81·121-s + 0.0894·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.358147182\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.358147182\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T - T^{2} + 2 T^{3} - p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | \( ( 1 - T )^{6} \) |
| good | 2 | \( 1 - p T^{2} + 7 T^{4} - 3 p^{2} T^{6} + 7 p^{2} T^{8} - p^{5} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 23 T^{2} + 307 T^{4} - 2586 T^{6} + 307 p^{2} T^{8} - 23 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 + T + 17 T^{2} + 10 T^{3} + 17 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 50 T^{2} + 1315 T^{4} - 21108 T^{6} + 1315 p^{2} T^{8} - 50 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 - 9 T + 19 T^{2} + 18 T^{3} + 19 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )( 1 + 9 T + 19 T^{2} - 18 T^{3} + 19 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} ) \) |
| 23 | \( 1 - 102 T^{2} + 4831 T^{4} - 138580 T^{6} + 4831 p^{2} T^{8} - 102 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 6 T + p T^{2} )^{6} \) |
| 31 | \( ( 1 + 37 T^{2} + 128 T^{3} + 37 p T^{4} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 166 T^{2} + 13011 T^{4} - 608316 T^{6} + 13011 p^{2} T^{8} - 166 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 6 T + 79 T^{2} + 516 T^{3} + 79 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 239 T^{2} + 24571 T^{4} - 1388154 T^{6} + 24571 p^{2} T^{8} - 239 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 95 T^{2} + 5443 T^{4} - 214314 T^{6} + 5443 p^{2} T^{8} - 95 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 162 T^{2} + 11539 T^{4} - 610708 T^{6} + 11539 p^{2} T^{8} - 162 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 - 10 T + 185 T^{2} - 1132 T^{3} + 185 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + 7 T + 79 T^{2} + 78 T^{3} + 79 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 62 T^{2} + 4771 T^{4} - 199788 T^{6} + 4771 p^{2} T^{8} - 62 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 26 T + 413 T^{2} + 4124 T^{3} + 413 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 307 T^{2} + 43299 T^{4} - 3822498 T^{6} + 43299 p^{2} T^{8} - 307 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 12 T + 229 T^{2} - 1864 T^{3} + 229 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 270 T^{2} + 39367 T^{4} - 3817060 T^{6} + 39367 p^{2} T^{8} - 270 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 - 12 T - 17 T^{2} + 1320 T^{3} - 17 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 554 T^{2} + 130507 T^{4} - 16717956 T^{6} + 130507 p^{2} T^{8} - 554 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−5.36387945295630377213066763438, −5.29521076006401585573314191413, −5.10482415866519286995933906796, −5.07795794433856567794343532271, −4.59587082209430712492227982087, −4.53401789975015927981378414551, −4.37151454589885872487660082684, −4.32289491246926012575726101773, −4.31798559317634937561398688010, −3.94943107913009062109312888964, −3.61866432742605536984680181024, −3.22589175694575022491115138366, −3.19544824594187826736925968366, −3.17644940240857250388479928074, −2.81015749332306697732178237401, −2.67431927381163926755411113721, −2.58514641659851077324678392742, −2.45633807964518455770823017436, −2.25204804917240107294739616917, −1.63804014048652847323990572895, −1.63373839324785596726698187873, −1.50544756630966692402952392224, −0.961246311443921929651030259647, −0.64763631516942824428922256579, −0.64216144947282649167307904214,
0.64216144947282649167307904214, 0.64763631516942824428922256579, 0.961246311443921929651030259647, 1.50544756630966692402952392224, 1.63373839324785596726698187873, 1.63804014048652847323990572895, 2.25204804917240107294739616917, 2.45633807964518455770823017436, 2.58514641659851077324678392742, 2.67431927381163926755411113721, 2.81015749332306697732178237401, 3.17644940240857250388479928074, 3.19544824594187826736925968366, 3.22589175694575022491115138366, 3.61866432742605536984680181024, 3.94943107913009062109312888964, 4.31798559317634937561398688010, 4.32289491246926012575726101773, 4.37151454589885872487660082684, 4.53401789975015927981378414551, 4.59587082209430712492227982087, 5.07795794433856567794343532271, 5.10482415866519286995933906796, 5.29521076006401585573314191413, 5.36387945295630377213066763438
Plot not available for L-functions of degree greater than 10.
