| L(s) = 1 | − 3-s − 2·4-s + 3·5-s + 7-s − 2·9-s + 11-s + 2·12-s − 4·13-s − 3·15-s + 4·16-s + 6·17-s − 2·19-s − 6·20-s − 21-s + 3·23-s + 4·25-s + 5·27-s − 2·28-s − 5·31-s − 33-s + 3·35-s + 4·36-s − 11·37-s + 4·39-s − 6·41-s − 8·43-s − 2·44-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 1.34·5-s + 0.377·7-s − 2/3·9-s + 0.301·11-s + 0.577·12-s − 1.10·13-s − 0.774·15-s + 16-s + 1.45·17-s − 0.458·19-s − 1.34·20-s − 0.218·21-s + 0.625·23-s + 4/5·25-s + 0.962·27-s − 0.377·28-s − 0.898·31-s − 0.174·33-s + 0.507·35-s + 2/3·36-s − 1.80·37-s + 0.640·39-s − 0.937·41-s − 1.21·43-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64757 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64757 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−14.36570758631727, −13.93505034261147, −13.70175505951947, −12.84198650981699, −12.57930479732431, −12.01838866822627, −11.57864789265739, −10.69911130189269, −10.44117516253771, −9.755679126477460, −9.576922480770022, −8.859132097156503, −8.475734935886061, −7.836694813702223, −7.148348721850879, −6.509894787551704, −5.931937853804475, −5.359177952203298, −5.053224316519039, −4.760593433441772, −3.521298880907077, −3.300569701419651, −2.228001361856921, −1.687269992810082, −0.8551027838320498, 0,
0.8551027838320498, 1.687269992810082, 2.228001361856921, 3.300569701419651, 3.521298880907077, 4.760593433441772, 5.053224316519039, 5.359177952203298, 5.931937853804475, 6.509894787551704, 7.148348721850879, 7.836694813702223, 8.475734935886061, 8.859132097156503, 9.576922480770022, 9.755679126477460, 10.44117516253771, 10.69911130189269, 11.57864789265739, 12.01838866822627, 12.57930479732431, 12.84198650981699, 13.70175505951947, 13.93505034261147, 14.36570758631727
