| L(s) = 1 | − 2.15·2-s + 2.63·4-s + 4.16·5-s − 1.36·8-s − 8.96·10-s + 0.785·11-s − 13-s − 2.32·16-s + 5.87·17-s + 8.59·19-s + 10.9·20-s − 1.69·22-s − 0.928·23-s + 12.3·25-s + 2.15·26-s + 5.23·29-s + 5.32·31-s + 7.74·32-s − 12.6·34-s − 1.27·37-s − 18.5·38-s − 5.69·40-s − 1.07·41-s + 8.59·43-s + 2.07·44-s + 2·46-s − 1.71·47-s + ⋯ |
| L(s) = 1 | − 1.52·2-s + 1.31·4-s + 1.86·5-s − 0.483·8-s − 2.83·10-s + 0.236·11-s − 0.277·13-s − 0.581·16-s + 1.42·17-s + 1.97·19-s + 2.45·20-s − 0.360·22-s − 0.193·23-s + 2.46·25-s + 0.422·26-s + 0.972·29-s + 0.956·31-s + 1.36·32-s − 2.16·34-s − 0.208·37-s − 3.00·38-s − 0.899·40-s − 0.167·41-s + 1.31·43-s + 0.312·44-s + 0.294·46-s − 0.250·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.680123166\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.680123166\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 2.15T + 2T^{2} \) |
| 5 | \( 1 - 4.16T + 5T^{2} \) |
| 11 | \( 1 - 0.785T + 11T^{2} \) |
| 17 | \( 1 - 5.87T + 17T^{2} \) |
| 19 | \( 1 - 8.59T + 19T^{2} \) |
| 23 | \( 1 + 0.928T + 23T^{2} \) |
| 29 | \( 1 - 5.23T + 29T^{2} \) |
| 31 | \( 1 - 5.32T + 31T^{2} \) |
| 37 | \( 1 + 1.27T + 37T^{2} \) |
| 41 | \( 1 + 1.07T + 41T^{2} \) |
| 43 | \( 1 - 8.59T + 43T^{2} \) |
| 47 | \( 1 + 1.71T + 47T^{2} \) |
| 53 | \( 1 + 6.80T + 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 0.729T + 67T^{2} \) |
| 71 | \( 1 - 7.53T + 71T^{2} \) |
| 73 | \( 1 - 7.32T + 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−8.108660329031819317209850002846, −7.65991779937655750842021328397, −6.78016965941968894055675615615, −6.17942091195579038973182923446, −5.42680564945270887438423071365, −4.76493175673768729743614238353, −3.18479695853119897252770857552, −2.50308405579077556137808624929, −1.44060311800018149276031212827, −1.01406378911669330729374296560,
1.01406378911669330729374296560, 1.44060311800018149276031212827, 2.50308405579077556137808624929, 3.18479695853119897252770857552, 4.76493175673768729743614238353, 5.42680564945270887438423071365, 6.17942091195579038973182923446, 6.78016965941968894055675615615, 7.65991779937655750842021328397, 8.108660329031819317209850002846
