| L(s) = 1 | + 2-s + 1.28·3-s + 4-s + 1.58·5-s + 1.28·6-s − 1.24·7-s + 8-s − 1.34·9-s + 1.58·10-s − 5.02·11-s + 1.28·12-s + 0.00314·13-s − 1.24·14-s + 2.04·15-s + 16-s − 5.35·17-s − 1.34·18-s − 5.89·19-s + 1.58·20-s − 1.59·21-s − 5.02·22-s − 1.67·23-s + 1.28·24-s − 2.47·25-s + 0.00314·26-s − 5.58·27-s − 1.24·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.742·3-s + 0.5·4-s + 0.710·5-s + 0.524·6-s − 0.469·7-s + 0.353·8-s − 0.448·9-s + 0.502·10-s − 1.51·11-s + 0.371·12-s + 0.000872·13-s − 0.332·14-s + 0.527·15-s + 0.250·16-s − 1.29·17-s − 0.317·18-s − 1.35·19-s + 0.355·20-s − 0.348·21-s − 1.07·22-s − 0.349·23-s + 0.262·24-s − 0.494·25-s + 0.000617·26-s − 1.07·27-s − 0.234·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2011 | \( 1 - T \) |
| good | 3 | \( 1 - 1.28T + 3T^{2} \) |
| 5 | \( 1 - 1.58T + 5T^{2} \) |
| 7 | \( 1 + 1.24T + 7T^{2} \) |
| 11 | \( 1 + 5.02T + 11T^{2} \) |
| 13 | \( 1 - 0.00314T + 13T^{2} \) |
| 17 | \( 1 + 5.35T + 17T^{2} \) |
| 19 | \( 1 + 5.89T + 19T^{2} \) |
| 23 | \( 1 + 1.67T + 23T^{2} \) |
| 29 | \( 1 - 2.22T + 29T^{2} \) |
| 31 | \( 1 + 3.72T + 31T^{2} \) |
| 37 | \( 1 - 7.31T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 + 0.426T + 43T^{2} \) |
| 47 | \( 1 + 3.67T + 47T^{2} \) |
| 53 | \( 1 + 9.13T + 53T^{2} \) |
| 59 | \( 1 + 4.81T + 59T^{2} \) |
| 61 | \( 1 + 0.550T + 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 + 0.384T + 71T^{2} \) |
| 73 | \( 1 + 2.57T + 73T^{2} \) |
| 79 | \( 1 + 8.48T + 79T^{2} \) |
| 83 | \( 1 + 3.59T + 83T^{2} \) |
| 89 | \( 1 - 9.29T + 89T^{2} \) |
| 97 | \( 1 - 17.5T + 97T^{2} \) |
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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
−8.048671108744159600732257726088, −7.41925814422969243722153747624, −6.17944970268223194314024224408, −6.10938269633666588822672283689, −5.01084901244885598314227958407, −4.27967016914173785558198435915, −3.27125235188601256166755833657, −2.42223817375273776119472851718, −2.10999434352427902304570597048, 0,
2.10999434352427902304570597048, 2.42223817375273776119472851718, 3.27125235188601256166755833657, 4.27967016914173785558198435915, 5.01084901244885598314227958407, 6.10938269633666588822672283689, 6.17944970268223194314024224408, 7.41925814422969243722153747624, 8.048671108744159600732257726088
