| L(s) = 1 | + 2.70·2-s − 1.18·3-s + 5.31·4-s + 5-s − 3.20·6-s + 3.05·7-s + 8.98·8-s − 1.59·9-s + 2.70·10-s − 3.40·11-s − 6.29·12-s + 4.10·13-s + 8.26·14-s − 1.18·15-s + 13.6·16-s + 1.33·17-s − 4.32·18-s − 2.30·19-s + 5.31·20-s − 3.61·21-s − 9.21·22-s − 10.6·24-s + 25-s + 11.1·26-s + 5.44·27-s + 16.2·28-s + 3.00·29-s + ⋯ |
| L(s) = 1 | + 1.91·2-s − 0.683·3-s + 2.65·4-s + 0.447·5-s − 1.30·6-s + 1.15·7-s + 3.17·8-s − 0.533·9-s + 0.855·10-s − 1.02·11-s − 1.81·12-s + 1.13·13-s + 2.20·14-s − 0.305·15-s + 3.41·16-s + 0.323·17-s − 1.01·18-s − 0.528·19-s + 1.18·20-s − 0.789·21-s − 1.96·22-s − 2.17·24-s + 0.200·25-s + 2.18·26-s + 1.04·27-s + 3.07·28-s + 0.558·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.068348401\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.068348401\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 2.70T + 2T^{2} \) |
| 3 | \( 1 + 1.18T + 3T^{2} \) |
| 7 | \( 1 - 3.05T + 7T^{2} \) |
| 11 | \( 1 + 3.40T + 11T^{2} \) |
| 13 | \( 1 - 4.10T + 13T^{2} \) |
| 17 | \( 1 - 1.33T + 17T^{2} \) |
| 19 | \( 1 + 2.30T + 19T^{2} \) |
| 29 | \( 1 - 3.00T + 29T^{2} \) |
| 31 | \( 1 - 8.33T + 31T^{2} \) |
| 37 | \( 1 + 9.37T + 37T^{2} \) |
| 41 | \( 1 - 1.68T + 41T^{2} \) |
| 43 | \( 1 + 5.83T + 43T^{2} \) |
| 47 | \( 1 + 9.13T + 47T^{2} \) |
| 53 | \( 1 - 1.54T + 53T^{2} \) |
| 59 | \( 1 - 0.536T + 59T^{2} \) |
| 61 | \( 1 - 4.54T + 61T^{2} \) |
| 67 | \( 1 - 9.36T + 67T^{2} \) |
| 71 | \( 1 - 9.54T + 71T^{2} \) |
| 73 | \( 1 + 8.53T + 73T^{2} \) |
| 79 | \( 1 - 1.38T + 79T^{2} \) |
| 83 | \( 1 - 5.75T + 83T^{2} \) |
| 89 | \( 1 + 7.08T + 89T^{2} \) |
| 97 | \( 1 + 7.71T + 97T^{2} \) |
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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
−8.394308695965706086354371484009, −8.072348200640178487766550846838, −6.77807194115627844683132397771, −6.32098939591240821001842663646, −5.36156777964711931747454431152, −5.20306202916727133536524507036, −4.35204038025405614876675515867, −3.28548980923872969786313523849, −2.42216586912202418256433731116, −1.39152944021538012897128845340,
1.39152944021538012897128845340, 2.42216586912202418256433731116, 3.28548980923872969786313523849, 4.35204038025405614876675515867, 5.20306202916727133536524507036, 5.36156777964711931747454431152, 6.32098939591240821001842663646, 6.77807194115627844683132397771, 8.072348200640178487766550846838, 8.394308695965706086354371484009
