| L(s) = 1 | − 5-s − 2·7-s − 11-s + 19-s + 25-s + 6·29-s − 2·31-s + 2·35-s − 8·37-s − 3·41-s − 4·43-s − 3·49-s + 6·53-s + 55-s − 9·59-s + 2·61-s + 13·67-s + 9·71-s − 5·73-s + 2·77-s + 5·79-s + 9·83-s − 95-s + 16·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s − 0.301·11-s + 0.229·19-s + 1/5·25-s + 1.11·29-s − 0.359·31-s + 0.338·35-s − 1.31·37-s − 0.468·41-s − 0.609·43-s − 3/7·49-s + 0.824·53-s + 0.134·55-s − 1.17·59-s + 0.256·61-s + 1.58·67-s + 1.06·71-s − 0.585·73-s + 0.227·77-s + 0.562·79-s + 0.987·83-s − 0.102·95-s + 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−12.75112949902674, −12.20209217718951, −12.12712000068680, −11.50374541087599, −10.85315981453445, −10.68470308680765, −9.938143478705032, −9.801663945735818, −9.140382313354368, −8.672701980548124, −8.217558985129282, −7.824027883305393, −7.157272424615231, −6.809953669483868, −6.400583566867145, −5.832106456594244, −5.144913053189162, −4.927761668529415, −4.216747300426989, −3.525768798484511, −3.380287749721129, −2.670221906286774, −2.115849620895330, −1.377937612263365, −0.6421256889044893, 0,
0.6421256889044893, 1.377937612263365, 2.115849620895330, 2.670221906286774, 3.380287749721129, 3.525768798484511, 4.216747300426989, 4.927761668529415, 5.144913053189162, 5.832106456594244, 6.400583566867145, 6.809953669483868, 7.157272424615231, 7.824027883305393, 8.217558985129282, 8.672701980548124, 9.140382313354368, 9.801663945735818, 9.938143478705032, 10.68470308680765, 10.85315981453445, 11.50374541087599, 12.12712000068680, 12.20209217718951, 12.75112949902674
