| L(s) = 1 | − 4·2-s + 6·3-s + 12·4-s + 2·5-s − 24·6-s + 28·7-s − 32·8-s + 27·9-s − 8·10-s − 2·11-s + 72·12-s + 48·13-s − 112·14-s + 12·15-s + 80·16-s − 4·17-s − 108·18-s + 78·19-s + 24·20-s + 168·21-s + 8·22-s + 46·23-s − 192·24-s + 30·25-s − 192·26-s + 108·27-s + 336·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.178·5-s − 1.63·6-s + 1.51·7-s − 1.41·8-s + 9-s − 0.252·10-s − 0.0548·11-s + 1.73·12-s + 1.02·13-s − 2.13·14-s + 0.206·15-s + 5/4·16-s − 0.0570·17-s − 1.41·18-s + 0.941·19-s + 0.268·20-s + 1.74·21-s + 0.0775·22-s + 0.417·23-s − 1.63·24-s + 6/25·25-s − 1.44·26-s + 0.769·27-s + 2.26·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19044 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19044 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.604240369\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.604240369\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 2 T - 26 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 p T + p^{3} T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 2386 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 48 T + 3862 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 8722 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 78 T + 14962 T^{2} - 78 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 184 T + 29542 T^{2} - 184 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 308 T + 82190 T^{2} - 308 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 10 p T + 133038 T^{2} - 10 p^{4} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 12 T - 21674 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 186 T + 154090 T^{2} - 186 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 528 T + 237454 T^{2} + 528 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 926 T + 505198 T^{2} + 926 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 396 T + p^{3} T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 42 T + 75190 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 434 T - 71862 T^{2} + 434 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 624 T + 454174 T^{2} - 624 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 352 T + 222878 T^{2} + 352 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 508 T + 979682 T^{2} + 508 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 994 T + 552658 T^{2} - 994 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 1050946 T^{2} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 784 T + 1791758 T^{2} + 784 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.92547374292640140061553706827, −12.43755053477478623304791088344, −11.51874829601357341761018361530, −11.46346895271236946855007320141, −10.80338180608618076843475459457, −10.32544424462697176954639539465, −9.599129657065987123088264412173, −9.377351173382723522284502619962, −8.663072910594084586633661286783, −8.198342416190642307891909845669, −7.912456106442472046486934570338, −7.56443054271149334091599031904, −6.50354488659117272555901927697, −6.24402351624511836170189210087, −4.97182291819632518984955449146, −4.48606376082978908015948317524, −3.22365923693788166579937934315, −2.69011601631536817334069759172, −1.54757530377360437695308602992, −1.13894015711225150596684799843,
1.13894015711225150596684799843, 1.54757530377360437695308602992, 2.69011601631536817334069759172, 3.22365923693788166579937934315, 4.48606376082978908015948317524, 4.97182291819632518984955449146, 6.24402351624511836170189210087, 6.50354488659117272555901927697, 7.56443054271149334091599031904, 7.912456106442472046486934570338, 8.198342416190642307891909845669, 8.663072910594084586633661286783, 9.377351173382723522284502619962, 9.599129657065987123088264412173, 10.32544424462697176954639539465, 10.80338180608618076843475459457, 11.46346895271236946855007320141, 11.51874829601357341761018361530, 12.43755053477478623304791088344, 12.92547374292640140061553706827
