| L(s) = 1 | − 1.41·3-s − 2.82·5-s − 25·9-s + 26·11-s − 33.9·13-s + 4.00·15-s + 103.·17-s + 94.7·19-s + 148·23-s − 117·25-s + 73.5·27-s − 118·29-s − 296.·31-s − 36.7·33-s − 254·37-s + 48·39-s − 91.9·41-s − 122·43-s + 70.7·45-s + 308.·47-s − 146·51-s − 170·53-s − 73.5·55-s − 134·57-s + 304.·59-s + 608.·61-s + 96·65-s + ⋯ |
| L(s) = 1 | − 0.272·3-s − 0.252·5-s − 0.925·9-s + 0.712·11-s − 0.724·13-s + 0.0688·15-s + 1.47·17-s + 1.14·19-s + 1.34·23-s − 0.936·25-s + 0.524·27-s − 0.755·29-s − 1.72·31-s − 0.193·33-s − 1.12·37-s + 0.197·39-s − 0.350·41-s − 0.432·43-s + 0.234·45-s + 0.956·47-s − 0.400·51-s − 0.440·53-s − 0.180·55-s − 0.311·57-s + 0.670·59-s + 1.27·61-s + 0.183·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 + 1.41T + 27T^{2} \) |
| 5 | \( 1 + 2.82T + 125T^{2} \) |
| 11 | \( 1 - 26T + 1.33e3T^{2} \) |
| 13 | \( 1 + 33.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 103.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 94.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 148T + 1.21e4T^{2} \) |
| 29 | \( 1 + 118T + 2.43e4T^{2} \) |
| 31 | \( 1 + 296.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 254T + 5.06e4T^{2} \) |
| 41 | \( 1 + 91.9T + 6.89e4T^{2} \) |
| 43 | \( 1 + 122T + 7.95e4T^{2} \) |
| 47 | \( 1 - 308.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 170T + 1.48e5T^{2} \) |
| 59 | \( 1 - 304.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 608.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 420T + 3.00e5T^{2} \) |
| 71 | \( 1 + 420T + 3.57e5T^{2} \) |
| 73 | \( 1 + 813.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.05e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.44e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.02e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 315.T + 9.12e5T^{2} \) |
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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
−9.442924578840818689829528617177, −8.728587819704367320950410560217, −7.59766865956455061953925001406, −7.03582443774509492737215025599, −5.65132402400041794933822493609, −5.27343105482806523939667259054, −3.79668786742900721425114941472, −2.96231705309785457058790353314, −1.39637029032782836671397570801, 0,
1.39637029032782836671397570801, 2.96231705309785457058790353314, 3.79668786742900721425114941472, 5.27343105482806523939667259054, 5.65132402400041794933822493609, 7.03582443774509492737215025599, 7.59766865956455061953925001406, 8.728587819704367320950410560217, 9.442924578840818689829528617177
