| L(s) = 1 | − 2.11·2-s + 1.11·3-s + 2.47·4-s − 1.35·5-s − 2.35·6-s − 3.11·7-s − 1.00·8-s − 1.75·9-s + 2.87·10-s − 3·11-s + 2.75·12-s + 3.94·13-s + 6.58·14-s − 1.51·15-s − 2.83·16-s − 7.47·17-s + 3.71·18-s + 2.47·19-s − 3.35·20-s − 3.47·21-s + 6.34·22-s + 1.41·23-s − 1.11·24-s − 3.15·25-s − 8.34·26-s − 5.30·27-s − 7.70·28-s + ⋯ |
| L(s) = 1 | − 1.49·2-s + 0.643·3-s + 1.23·4-s − 0.607·5-s − 0.962·6-s − 1.17·7-s − 0.353·8-s − 0.585·9-s + 0.908·10-s − 0.904·11-s + 0.795·12-s + 1.09·13-s + 1.76·14-s − 0.390·15-s − 0.707·16-s − 1.81·17-s + 0.875·18-s + 0.567·19-s − 0.750·20-s − 0.757·21-s + 1.35·22-s + 0.294·23-s − 0.227·24-s − 0.631·25-s − 1.63·26-s − 1.02·27-s − 1.45·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 211 | \( 1 + T \) |
| good | 2 | \( 1 + 2.11T + 2T^{2} \) |
| 3 | \( 1 - 1.11T + 3T^{2} \) |
| 5 | \( 1 + 1.35T + 5T^{2} \) |
| 7 | \( 1 + 3.11T + 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - 3.94T + 13T^{2} \) |
| 17 | \( 1 + 7.47T + 17T^{2} \) |
| 19 | \( 1 - 2.47T + 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 + 10.5T + 29T^{2} \) |
| 31 | \( 1 + 4.41T + 31T^{2} \) |
| 37 | \( 1 - 5.58T + 37T^{2} \) |
| 41 | \( 1 - 8.10T + 41T^{2} \) |
| 43 | \( 1 - 9.45T + 43T^{2} \) |
| 47 | \( 1 + 3.75T + 47T^{2} \) |
| 53 | \( 1 - 1.47T + 53T^{2} \) |
| 59 | \( 1 - 3.53T + 59T^{2} \) |
| 61 | \( 1 - 0.926T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 5.21T + 71T^{2} \) |
| 73 | \( 1 - 3.11T + 73T^{2} \) |
| 79 | \( 1 + 7.96T + 79T^{2} \) |
| 83 | \( 1 - 17.1T + 83T^{2} \) |
| 89 | \( 1 + 5.91T + 89T^{2} \) |
| 97 | \( 1 - 1.45T + 97T^{2} \) |
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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
−11.29445540204577409493198252830, −10.89108290853046440875698180983, −9.466567757847061403970723753950, −9.048661137321365576864515735104, −8.055753645423318784752524598922, −7.26871587572049958285890867439, −5.97452654890392450463764696467, −3.83581175539979428399937321229, −2.45169575634776740716685923531, 0,
2.45169575634776740716685923531, 3.83581175539979428399937321229, 5.97452654890392450463764696467, 7.26871587572049958285890867439, 8.055753645423318784752524598922, 9.048661137321365576864515735104, 9.466567757847061403970723753950, 10.89108290853046440875698180983, 11.29445540204577409493198252830
