| L(s) = 1 | + 1.60·2-s + 0.427·3-s + 0.574·4-s + 1.60·5-s + 0.686·6-s + 0.796·7-s − 2.28·8-s − 2.81·9-s + 2.57·10-s − 5.12·11-s + 0.245·12-s + 0.169·13-s + 1.27·14-s + 0.685·15-s − 4.81·16-s + 17-s − 4.51·18-s − 8.09·19-s + 0.920·20-s + 0.340·21-s − 8.22·22-s + 7.76·23-s − 0.978·24-s − 2.43·25-s + 0.271·26-s − 2.48·27-s + 0.457·28-s + ⋯ |
| L(s) = 1 | + 1.13·2-s + 0.247·3-s + 0.287·4-s + 0.716·5-s + 0.280·6-s + 0.300·7-s − 0.808·8-s − 0.938·9-s + 0.813·10-s − 1.54·11-s + 0.0709·12-s + 0.0468·13-s + 0.341·14-s + 0.177·15-s − 1.20·16-s + 0.242·17-s − 1.06·18-s − 1.85·19-s + 0.205·20-s + 0.0743·21-s − 1.75·22-s + 1.61·23-s − 0.199·24-s − 0.486·25-s + 0.0531·26-s − 0.479·27-s + 0.0863·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2669 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2669 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 - T \) |
| 157 | \( 1 - T \) |
| good | 2 | \( 1 - 1.60T + 2T^{2} \) |
| 3 | \( 1 - 0.427T + 3T^{2} \) |
| 5 | \( 1 - 1.60T + 5T^{2} \) |
| 7 | \( 1 - 0.796T + 7T^{2} \) |
| 11 | \( 1 + 5.12T + 11T^{2} \) |
| 13 | \( 1 - 0.169T + 13T^{2} \) |
| 19 | \( 1 + 8.09T + 19T^{2} \) |
| 23 | \( 1 - 7.76T + 23T^{2} \) |
| 29 | \( 1 - 2.62T + 29T^{2} \) |
| 31 | \( 1 + 3.99T + 31T^{2} \) |
| 37 | \( 1 - 5.69T + 37T^{2} \) |
| 41 | \( 1 + 9.29T + 41T^{2} \) |
| 43 | \( 1 + 0.754T + 43T^{2} \) |
| 47 | \( 1 + 2.98T + 47T^{2} \) |
| 53 | \( 1 + 9.99T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 4.19T + 61T^{2} \) |
| 67 | \( 1 - 6.52T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 - 1.86T + 79T^{2} \) |
| 83 | \( 1 - 2.25T + 83T^{2} \) |
| 89 | \( 1 - 2.67T + 89T^{2} \) |
| 97 | \( 1 - 2.24T + 97T^{2} \) |
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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
−8.435618248113947937115990851820, −7.82712732791804843457859547963, −6.53402450191910261585709674954, −6.02901297097525710185600532520, −5.07112761191732593258113030141, −4.87565003194135476019234043651, −3.54590646264911122549842820919, −2.78948858596095007401702944288, −2.05932635036446012770256120061, 0,
2.05932635036446012770256120061, 2.78948858596095007401702944288, 3.54590646264911122549842820919, 4.87565003194135476019234043651, 5.07112761191732593258113030141, 6.02901297097525710185600532520, 6.53402450191910261585709674954, 7.82712732791804843457859547963, 8.435618248113947937115990851820
