| L(s) = 1 | − 2·2-s − 7·3-s + 4·4-s − 19·5-s + 14·6-s + 14·7-s − 8·8-s + 22·9-s + 38·10-s + 11·11-s − 28·12-s − 72·13-s − 28·14-s + 133·15-s + 16·16-s − 46·17-s − 44·18-s − 20·19-s − 76·20-s − 98·21-s − 22·22-s − 107·23-s + 56·24-s + 236·25-s + 144·26-s + 35·27-s + 56·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.34·3-s + 1/2·4-s − 1.69·5-s + 0.952·6-s + 0.755·7-s − 0.353·8-s + 0.814·9-s + 1.20·10-s + 0.301·11-s − 0.673·12-s − 1.53·13-s − 0.534·14-s + 2.28·15-s + 1/4·16-s − 0.656·17-s − 0.576·18-s − 0.241·19-s − 0.849·20-s − 1.01·21-s − 0.213·22-s − 0.970·23-s + 0.476·24-s + 1.88·25-s + 1.08·26-s + 0.249·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 11 | \( 1 - p T \) |
| good | 3 | \( 1 + 7 T + p^{3} T^{2} \) |
| 5 | \( 1 + 19 T + p^{3} T^{2} \) |
| 7 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 13 | \( 1 + 72 T + p^{3} T^{2} \) |
| 17 | \( 1 + 46 T + p^{3} T^{2} \) |
| 19 | \( 1 + 20 T + p^{3} T^{2} \) |
| 23 | \( 1 + 107 T + p^{3} T^{2} \) |
| 29 | \( 1 - 120 T + p^{3} T^{2} \) |
| 31 | \( 1 - 117 T + p^{3} T^{2} \) |
| 37 | \( 1 + 201 T + p^{3} T^{2} \) |
| 41 | \( 1 + 228 T + p^{3} T^{2} \) |
| 43 | \( 1 + 242 T + p^{3} T^{2} \) |
| 47 | \( 1 + 96 T + p^{3} T^{2} \) |
| 53 | \( 1 - 458 T + p^{3} T^{2} \) |
| 59 | \( 1 - 435 T + p^{3} T^{2} \) |
| 61 | \( 1 + 668 T + p^{3} T^{2} \) |
| 67 | \( 1 - 439 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1113 T + p^{3} T^{2} \) |
| 73 | \( 1 + 72 T + p^{3} T^{2} \) |
| 79 | \( 1 + 70 T + p^{3} T^{2} \) |
| 83 | \( 1 - 358 T + p^{3} T^{2} \) |
| 89 | \( 1 - 895 T + p^{3} T^{2} \) |
| 97 | \( 1 - 409 T + p^{3} T^{2} \) |
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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
−17.02460814822688922423139274942, −15.96668702533574902162305414750, −14.85556452217872291262912762142, −12.02880401333640069826073952224, −11.73521762301427298260050259981, −10.45866853395596421580186603791, −8.254955748278688089446281627149, −6.94002866642875078725411060341, −4.70853827720181835846537287104, 0,
4.70853827720181835846537287104, 6.94002866642875078725411060341, 8.254955748278688089446281627149, 10.45866853395596421580186603791, 11.73521762301427298260050259981, 12.02880401333640069826073952224, 14.85556452217872291262912762142, 15.96668702533574902162305414750, 17.02460814822688922423139274942
