L(s) = 1 | + 0.488·2-s + 3-s − 1.76·4-s + 0.529·5-s + 0.488·6-s − 1.83·8-s + 9-s + 0.258·10-s − 1.22·11-s − 1.76·12-s + 1.15·13-s + 0.529·15-s + 2.62·16-s − 6.79·17-s + 0.488·18-s + 2.34·19-s − 0.933·20-s − 0.599·22-s + 23-s − 1.83·24-s − 4.71·25-s + 0.561·26-s + 27-s − 5.30·29-s + 0.258·30-s + 10.3·31-s + 4.95·32-s + ⋯ |
L(s) = 1 | + 0.345·2-s + 0.577·3-s − 0.880·4-s + 0.236·5-s + 0.199·6-s − 0.649·8-s + 0.333·9-s + 0.0817·10-s − 0.370·11-s − 0.508·12-s + 0.319·13-s + 0.136·15-s + 0.656·16-s − 1.64·17-s + 0.115·18-s + 0.537·19-s − 0.208·20-s − 0.127·22-s + 0.208·23-s − 0.374·24-s − 0.943·25-s + 0.110·26-s + 0.192·27-s − 0.984·29-s + 0.0472·30-s + 1.85·31-s + 0.875·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 - 0.488T + 2T^{2} \) |
| 5 | \( 1 - 0.529T + 5T^{2} \) |
| 11 | \( 1 + 1.22T + 11T^{2} \) |
| 13 | \( 1 - 1.15T + 13T^{2} \) |
| 17 | \( 1 + 6.79T + 17T^{2} \) |
| 19 | \( 1 - 2.34T + 19T^{2} \) |
| 29 | \( 1 + 5.30T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 - 1.58T + 37T^{2} \) |
| 41 | \( 1 - 6.36T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 9.66T + 61T^{2} \) |
| 67 | \( 1 - 15.1T + 67T^{2} \) |
| 71 | \( 1 + 7.49T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 - 7.21T + 89T^{2} \) |
| 97 | \( 1 + 8.00T + 97T^{2} \) |
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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
−8.241717613842556569505610784536, −7.75728266532011323972205557774, −6.58992837655666860359432279876, −6.00016886754608703449026259762, −4.92524337754754993951713354322, −4.46052415418204641397135530896, −3.51462771717133468585385272633, −2.72956104812659916080441465036, −1.57786710814096676856157246516, 0,
1.57786710814096676856157246516, 2.72956104812659916080441465036, 3.51462771717133468585385272633, 4.46052415418204641397135530896, 4.92524337754754993951713354322, 6.00016886754608703449026259762, 6.58992837655666860359432279876, 7.75728266532011323972205557774, 8.241717613842556569505610784536