| L(s) = 1 | − 2-s + 2.37·3-s + 4-s + 5-s − 2.37·6-s − 2.37·7-s − 8-s + 2.62·9-s − 10-s − 11-s + 2.37·12-s + 2·13-s + 2.37·14-s + 2.37·15-s + 16-s − 4.37·17-s − 2.62·18-s + 6.37·19-s + 20-s − 5.62·21-s + 22-s − 8.74·23-s − 2.37·24-s + 25-s − 2·26-s − 0.883·27-s − 2.37·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.36·3-s + 0.5·4-s + 0.447·5-s − 0.968·6-s − 0.896·7-s − 0.353·8-s + 0.875·9-s − 0.316·10-s − 0.301·11-s + 0.684·12-s + 0.554·13-s + 0.634·14-s + 0.612·15-s + 0.250·16-s − 1.06·17-s − 0.619·18-s + 1.46·19-s + 0.223·20-s − 1.22·21-s + 0.213·22-s − 1.82·23-s − 0.484·24-s + 0.200·25-s − 0.392·26-s − 0.169·27-s − 0.448·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.092978502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.092978502\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 2.37T + 3T^{2} \) |
| 7 | \( 1 + 2.37T + 7T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 4.37T + 17T^{2} \) |
| 19 | \( 1 - 6.37T + 19T^{2} \) |
| 23 | \( 1 + 8.74T + 23T^{2} \) |
| 29 | \( 1 + 4.37T + 29T^{2} \) |
| 31 | \( 1 + 2.37T + 31T^{2} \) |
| 37 | \( 1 - 3.62T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 8.74T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 - 8.74T + 59T^{2} \) |
| 61 | \( 1 - 0.372T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 7.11T + 71T^{2} \) |
| 73 | \( 1 - 7.48T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 - 8.74T + 83T^{2} \) |
| 89 | \( 1 - 4.37T + 89T^{2} \) |
| 97 | \( 1 + 1.25T + 97T^{2} \) |
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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
−13.65791280221585508799586150832, −12.95801060937520669818156462953, −11.42377555288379445698982492618, −9.979494288438638459727049279184, −9.390247785900182046123628465206, −8.415837047933552989842722452006, −7.35422687001489073777346203906, −5.98353161137008289164431147583, −3.61519141027645554829414607861, −2.28619853611828694232914044186,
2.28619853611828694232914044186, 3.61519141027645554829414607861, 5.98353161137008289164431147583, 7.35422687001489073777346203906, 8.415837047933552989842722452006, 9.390247785900182046123628465206, 9.979494288438638459727049279184, 11.42377555288379445698982492618, 12.95801060937520669818156462953, 13.65791280221585508799586150832
