| L(s) = 1 | + 1.99·2-s + 2.01·3-s + 1.96·4-s + 0.195·5-s + 4.01·6-s − 1.27·7-s − 0.0674·8-s + 1.05·9-s + 0.388·10-s − 5.49·11-s + 3.95·12-s − 2.53·14-s + 0.392·15-s − 4.06·16-s − 5.47·17-s + 2.10·18-s + 2.83·19-s + 0.383·20-s − 2.55·21-s − 10.9·22-s − 5.48·23-s − 0.135·24-s − 4.96·25-s − 3.91·27-s − 2.49·28-s + 9.37·29-s + 0.782·30-s + ⋯ |
| L(s) = 1 | + 1.40·2-s + 1.16·3-s + 0.983·4-s + 0.0872·5-s + 1.63·6-s − 0.480·7-s − 0.0238·8-s + 0.351·9-s + 0.122·10-s − 1.65·11-s + 1.14·12-s − 0.676·14-s + 0.101·15-s − 1.01·16-s − 1.32·17-s + 0.495·18-s + 0.650·19-s + 0.0857·20-s − 0.558·21-s − 2.33·22-s − 1.14·23-s − 0.0277·24-s − 0.992·25-s − 0.753·27-s − 0.472·28-s + 1.74·29-s + 0.142·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 1.99T + 2T^{2} \) |
| 3 | \( 1 - 2.01T + 3T^{2} \) |
| 5 | \( 1 - 0.195T + 5T^{2} \) |
| 7 | \( 1 + 1.27T + 7T^{2} \) |
| 11 | \( 1 + 5.49T + 11T^{2} \) |
| 17 | \( 1 + 5.47T + 17T^{2} \) |
| 19 | \( 1 - 2.83T + 19T^{2} \) |
| 23 | \( 1 + 5.48T + 23T^{2} \) |
| 29 | \( 1 - 9.37T + 29T^{2} \) |
| 37 | \( 1 - 4.25T + 37T^{2} \) |
| 41 | \( 1 - 7.13T + 41T^{2} \) |
| 43 | \( 1 - 3.89T + 43T^{2} \) |
| 47 | \( 1 + 1.85T + 47T^{2} \) |
| 53 | \( 1 + 9.40T + 53T^{2} \) |
| 59 | \( 1 - 2.77T + 59T^{2} \) |
| 61 | \( 1 + 6.42T + 61T^{2} \) |
| 67 | \( 1 + 9.63T + 67T^{2} \) |
| 71 | \( 1 - 15.5T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 + 15.9T + 79T^{2} \) |
| 83 | \( 1 - 5.07T + 83T^{2} \) |
| 89 | \( 1 + 4.65T + 89T^{2} \) |
| 97 | \( 1 + 9.50T + 97T^{2} \) |
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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
−8.009251568334485160879071212567, −7.02332662473012290591008600230, −6.17764784022006963857126465530, −5.62661191435894376591820061590, −4.71153623609664183899656979351, −4.12449688577298115176414122612, −3.20746264605630087703283721883, −2.67155854165795566436855228292, −2.12544985960694890430382202346, 0,
2.12544985960694890430382202346, 2.67155854165795566436855228292, 3.20746264605630087703283721883, 4.12449688577298115176414122612, 4.71153623609664183899656979351, 5.62661191435894376591820061590, 6.17764784022006963857126465530, 7.02332662473012290591008600230, 8.009251568334485160879071212567
