| L(s) = 1 | + 2·5-s − 7-s + 4·11-s − 2·13-s − 2·17-s + 4·19-s − 25-s − 29-s + 8·31-s − 2·35-s − 10·37-s + 6·41-s − 12·43-s − 8·47-s + 49-s − 6·53-s + 8·55-s + 12·59-s − 10·61-s − 4·65-s + 12·67-s − 16·71-s + 2·73-s − 4·77-s + 4·83-s − 4·85-s + 6·89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.377·7-s + 1.20·11-s − 0.554·13-s − 0.485·17-s + 0.917·19-s − 1/5·25-s − 0.185·29-s + 1.43·31-s − 0.338·35-s − 1.64·37-s + 0.937·41-s − 1.82·43-s − 1.16·47-s + 1/7·49-s − 0.824·53-s + 1.07·55-s + 1.56·59-s − 1.28·61-s − 0.496·65-s + 1.46·67-s − 1.89·71-s + 0.234·73-s − 0.455·77-s + 0.439·83-s − 0.433·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.50933135071868, −14.68416612686554, −14.45573038478328, −13.67685231590693, −13.52751455971721, −12.86907743424624, −12.06383705281208, −11.86258527364503, −11.24040153686336, −10.42302965671897, −9.944969481846247, −9.520713933025390, −9.096599641549816, −8.401839167703418, −7.779720152059534, −6.865274839875722, −6.658971243557430, −6.038605298543407, −5.312427084785723, −4.809053694983537, −3.988758767538846, −3.327549130114378, −2.626255984702195, −1.795481161979592, −1.212530555274480, 0,
1.212530555274480, 1.795481161979592, 2.626255984702195, 3.327549130114378, 3.988758767538846, 4.809053694983537, 5.312427084785723, 6.038605298543407, 6.658971243557430, 6.865274839875722, 7.779720152059534, 8.401839167703418, 9.096599641549816, 9.520713933025390, 9.944969481846247, 10.42302965671897, 11.24040153686336, 11.86258527364503, 12.06383705281208, 12.86907743424624, 13.52751455971721, 13.67685231590693, 14.45573038478328, 14.68416612686554, 15.50933135071868
