| L(s) = 1 | + 2.18·3-s + 2.34·5-s + 7·7-s − 22.2·9-s − 11·11-s − 21.0·13-s + 5.12·15-s + 86.0·17-s − 129.·19-s + 15.3·21-s + 109.·23-s − 119.·25-s − 107.·27-s − 149.·29-s − 284.·31-s − 24.0·33-s + 16.4·35-s − 107.·37-s − 46.0·39-s + 505.·41-s + 26.8·43-s − 52.0·45-s − 12.1·47-s + 49·49-s + 188.·51-s − 571.·53-s − 25.7·55-s + ⋯ |
| L(s) = 1 | + 0.421·3-s + 0.209·5-s + 0.377·7-s − 0.822·9-s − 0.301·11-s − 0.449·13-s + 0.0882·15-s + 1.22·17-s − 1.56·19-s + 0.159·21-s + 0.992·23-s − 0.956·25-s − 0.767·27-s − 0.957·29-s − 1.64·31-s − 0.126·33-s + 0.0792·35-s − 0.476·37-s − 0.189·39-s + 1.92·41-s + 0.0951·43-s − 0.172·45-s − 0.0377·47-s + 0.142·49-s + 0.517·51-s − 1.48·53-s − 0.0631·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 + 11T \) |
| good | 3 | \( 1 - 2.18T + 27T^{2} \) |
| 5 | \( 1 - 2.34T + 125T^{2} \) |
| 13 | \( 1 + 21.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 86.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 129.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 109.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 149.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 284.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 107.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 505.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 26.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + 12.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + 571.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 53.0T + 2.05e5T^{2} \) |
| 61 | \( 1 + 469.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 732.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 338.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.14e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 156.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 960.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 466.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 126.T + 9.12e5T^{2} \) |
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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
−9.640604134880118586568732573137, −8.946982887769362984662835010103, −8.014520242179649609807461754021, −7.34252774801274780904707278729, −5.98356511243440000664740212127, −5.29250387270163217201251430345, −4.00504752181082623528343567728, −2.86312927513021288933869135879, −1.79332899796971248873645869926, 0,
1.79332899796971248873645869926, 2.86312927513021288933869135879, 4.00504752181082623528343567728, 5.29250387270163217201251430345, 5.98356511243440000664740212127, 7.34252774801274780904707278729, 8.014520242179649609807461754021, 8.946982887769362984662835010103, 9.640604134880118586568732573137
