| L(s) = 1 | − 2.26·2-s + 3.11·4-s + 5-s + 1.26·7-s − 2.52·8-s − 2.26·10-s + 4.52·11-s − 2.85·14-s − 0.523·16-s + 4.49·17-s − 5.11·19-s + 3.11·20-s − 10.2·22-s − 2.23·23-s + 25-s + 3.93·28-s + 1.37·29-s − 8.87·31-s + 6.23·32-s − 10.1·34-s + 1.26·35-s + 0.231·37-s + 11.5·38-s − 2.52·40-s + 1.14·41-s + 6.37·43-s + 14.0·44-s + ⋯ |
| L(s) = 1 | − 1.59·2-s + 1.55·4-s + 0.447·5-s + 0.476·7-s − 0.892·8-s − 0.715·10-s + 1.36·11-s − 0.762·14-s − 0.130·16-s + 1.08·17-s − 1.17·19-s + 0.696·20-s − 2.18·22-s − 0.465·23-s + 0.200·25-s + 0.742·28-s + 0.255·29-s − 1.59·31-s + 1.10·32-s − 1.74·34-s + 0.213·35-s + 0.0380·37-s + 1.87·38-s − 0.399·40-s + 0.178·41-s + 0.972·43-s + 2.12·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.26T + 2T^{2} \) |
| 7 | \( 1 - 1.26T + 7T^{2} \) |
| 11 | \( 1 - 4.52T + 11T^{2} \) |
| 17 | \( 1 - 4.49T + 17T^{2} \) |
| 19 | \( 1 + 5.11T + 19T^{2} \) |
| 23 | \( 1 + 2.23T + 23T^{2} \) |
| 29 | \( 1 - 1.37T + 29T^{2} \) |
| 31 | \( 1 + 8.87T + 31T^{2} \) |
| 37 | \( 1 - 0.231T + 37T^{2} \) |
| 41 | \( 1 - 1.14T + 41T^{2} \) |
| 43 | \( 1 - 6.37T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 + 4.52T + 53T^{2} \) |
| 59 | \( 1 + 0.853T + 59T^{2} \) |
| 61 | \( 1 - 4.63T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 + 9.60T + 71T^{2} \) |
| 73 | \( 1 + 13.7T + 73T^{2} \) |
| 79 | \( 1 + 8.87T + 79T^{2} \) |
| 83 | \( 1 + 8.23T + 83T^{2} \) |
| 89 | \( 1 - 6.62T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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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
−7.70231887915285523439777515849, −7.04910265352168294128889649272, −6.34252055364059662500374024090, −5.76111336381996644627705596445, −4.66055279659388285275401737230, −3.87826655564846433774822227088, −2.78783314704637761110866440851, −1.65912598273195686657349874909, −1.40428884928330874270402697359, 0,
1.40428884928330874270402697359, 1.65912598273195686657349874909, 2.78783314704637761110866440851, 3.87826655564846433774822227088, 4.66055279659388285275401737230, 5.76111336381996644627705596445, 6.34252055364059662500374024090, 7.04910265352168294128889649272, 7.70231887915285523439777515849
