| L(s) = 1 | + 2.62·3-s − 0.341·5-s + 3.87·9-s − 1.63·11-s − 3.79·13-s − 0.895·15-s − 0.563·17-s + 0.326·19-s − 6.42·23-s − 4.88·25-s + 2.30·27-s − 2.13·29-s − 2.42·31-s − 4.28·33-s + 9.94·37-s − 9.95·39-s − 7.60·41-s + 3.48·43-s − 1.32·45-s − 7.79·47-s − 1.47·51-s − 7.82·53-s + 0.558·55-s + 0.857·57-s − 1.63·59-s + 0.163·61-s + 1.29·65-s + ⋯ |
| L(s) = 1 | + 1.51·3-s − 0.152·5-s + 1.29·9-s − 0.492·11-s − 1.05·13-s − 0.231·15-s − 0.136·17-s + 0.0750·19-s − 1.33·23-s − 0.976·25-s + 0.443·27-s − 0.396·29-s − 0.436·31-s − 0.746·33-s + 1.63·37-s − 1.59·39-s − 1.18·41-s + 0.532·43-s − 0.197·45-s − 1.13·47-s − 0.206·51-s − 1.07·53-s + 0.0752·55-s + 0.113·57-s − 0.213·59-s + 0.0209·61-s + 0.160·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2.62T + 3T^{2} \) |
| 5 | \( 1 + 0.341T + 5T^{2} \) |
| 11 | \( 1 + 1.63T + 11T^{2} \) |
| 13 | \( 1 + 3.79T + 13T^{2} \) |
| 17 | \( 1 + 0.563T + 17T^{2} \) |
| 19 | \( 1 - 0.326T + 19T^{2} \) |
| 23 | \( 1 + 6.42T + 23T^{2} \) |
| 29 | \( 1 + 2.13T + 29T^{2} \) |
| 31 | \( 1 + 2.42T + 31T^{2} \) |
| 37 | \( 1 - 9.94T + 37T^{2} \) |
| 41 | \( 1 + 7.60T + 41T^{2} \) |
| 43 | \( 1 - 3.48T + 43T^{2} \) |
| 47 | \( 1 + 7.79T + 47T^{2} \) |
| 53 | \( 1 + 7.82T + 53T^{2} \) |
| 59 | \( 1 + 1.63T + 59T^{2} \) |
| 61 | \( 1 - 0.163T + 61T^{2} \) |
| 67 | \( 1 + 5.08T + 67T^{2} \) |
| 71 | \( 1 - 2.80T + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 - 5.19T + 97T^{2} \) |
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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
−7.81202156912240643808911259013, −7.51107021221269965842462798982, −6.50875658958939891042308594860, −5.61466605155258136336429233612, −4.66575397018139077072260227591, −3.95007556088843293703668703560, −3.17977535441913716378030700514, −2.39461415429406691070448001000, −1.77188605288945043445793576907, 0,
1.77188605288945043445793576907, 2.39461415429406691070448001000, 3.17977535441913716378030700514, 3.95007556088843293703668703560, 4.66575397018139077072260227591, 5.61466605155258136336429233612, 6.50875658958939891042308594860, 7.51107021221269965842462798982, 7.81202156912240643808911259013
