| L(s) = 1 | + 3·2-s + 6·4-s − 2·7-s + 10·8-s − 2·11-s + 6·13-s − 6·14-s + 15·16-s + 12·17-s + 3·19-s − 6·22-s − 2·23-s + 18·26-s − 12·28-s − 6·29-s + 8·31-s + 21·32-s + 36·34-s + 37-s + 9·38-s + 14·43-s − 12·44-s − 6·46-s + 3·47-s − 4·49-s + 36·52-s − 10·53-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3·4-s − 0.755·7-s + 3.53·8-s − 0.603·11-s + 1.66·13-s − 1.60·14-s + 15/4·16-s + 2.91·17-s + 0.688·19-s − 1.27·22-s − 0.417·23-s + 3.53·26-s − 2.26·28-s − 1.11·29-s + 1.43·31-s + 3.71·32-s + 6.17·34-s + 0.164·37-s + 1.45·38-s + 2.13·43-s − 1.80·44-s − 0.884·46-s + 0.437·47-s − 4/7·49-s + 4.99·52-s − 1.37·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(31.97613046\) |
| \(L(\frac12)\) |
\(\approx\) |
\(31.97613046\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{3} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + 2 T + 8 T^{2} + 29 T^{3} + 8 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 2 T + 9 T^{2} + 20 T^{3} + 9 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 6 T + 36 T^{2} - 121 T^{3} + 36 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 12 T + 84 T^{2} - 399 T^{3} + 84 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 2 T + 18 T^{2} + 203 T^{3} + 18 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 6 T + 84 T^{2} + 339 T^{3} + 84 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 8 T + 89 T^{2} - 440 T^{3} + 89 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - T + 86 T^{2} - 73 T^{3} + 86 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 43 | $S_4\times C_2$ | \( 1 - 14 T + 169 T^{2} - 1180 T^{3} + 169 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 3 T + 84 T^{2} - 327 T^{3} + 84 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 10 T + 54 T^{2} + 193 T^{3} + 54 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 6 T - 24 T^{2} - 723 T^{3} - 24 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 2 T + 127 T^{2} + 124 T^{3} + 127 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 4 T + 178 T^{2} + 461 T^{3} + 178 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 6 T + 21 T^{2} + 660 T^{3} + 21 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 12 T + 192 T^{2} - 1435 T^{3} + 192 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 20 T + 269 T^{2} - 2840 T^{3} + 269 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 4 T + 141 T^{2} + 832 T^{3} + 141 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 14 T + 231 T^{2} + 2180 T^{3} + 231 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 28 T + 335 T^{2} - 2992 T^{3} + 335 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.79533541352732307153126256522, −6.40539119469154462206386596287, −6.28722161843378607534484918352, −6.07518533346252307982447150105, −5.79204253632257805217041888646, −5.71671546062902157068707269324, −5.69703420703230550869350010575, −5.09197316285553652745093095958, −5.03164312509743599088734291232, −4.89826441990342736468613418082, −4.51904907102634740694638638841, −4.13435083673753574621056004672, −3.96695942809167238035922419254, −3.69069985167752475854001438757, −3.50313518134241714404599788574, −3.36538745545436308637899757374, −3.01405016741945134544264615291, −2.84441873765008972394768517251, −2.68561167526379193647703290328, −2.15341740744582934154275570692, −1.85482429620308785979675212446, −1.57529356337536740220761332011, −1.10089495407072908866481699028, −0.907594394310497381305841206152, −0.53927409296349439025322965091,
0.53927409296349439025322965091, 0.907594394310497381305841206152, 1.10089495407072908866481699028, 1.57529356337536740220761332011, 1.85482429620308785979675212446, 2.15341740744582934154275570692, 2.68561167526379193647703290328, 2.84441873765008972394768517251, 3.01405016741945134544264615291, 3.36538745545436308637899757374, 3.50313518134241714404599788574, 3.69069985167752475854001438757, 3.96695942809167238035922419254, 4.13435083673753574621056004672, 4.51904907102634740694638638841, 4.89826441990342736468613418082, 5.03164312509743599088734291232, 5.09197316285553652745093095958, 5.69703420703230550869350010575, 5.71671546062902157068707269324, 5.79204253632257805217041888646, 6.07518533346252307982447150105, 6.28722161843378607534484918352, 6.40539119469154462206386596287, 6.79533541352732307153126256522
