| L(s) = 1 | + 2-s − 1.65·3-s + 4-s − 2.90·5-s − 1.65·6-s + 0.175·7-s + 8-s − 0.273·9-s − 2.90·10-s + 4.59·11-s − 1.65·12-s + 2.27·13-s + 0.175·14-s + 4.80·15-s + 16-s − 7.35·17-s − 0.273·18-s − 1.09·19-s − 2.90·20-s − 0.289·21-s + 4.59·22-s − 1.49·23-s − 1.65·24-s + 3.45·25-s + 2.27·26-s + 5.40·27-s + 0.175·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.953·3-s + 0.5·4-s − 1.30·5-s − 0.674·6-s + 0.0661·7-s + 0.353·8-s − 0.0911·9-s − 0.919·10-s + 1.38·11-s − 0.476·12-s + 0.630·13-s + 0.0467·14-s + 1.24·15-s + 0.250·16-s − 1.78·17-s − 0.0644·18-s − 0.252·19-s − 0.650·20-s − 0.0630·21-s + 0.978·22-s − 0.312·23-s − 0.337·24-s + 0.691·25-s + 0.446·26-s + 1.04·27-s + 0.0330·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 + 1.65T + 3T^{2} \) |
| 5 | \( 1 + 2.90T + 5T^{2} \) |
| 7 | \( 1 - 0.175T + 7T^{2} \) |
| 11 | \( 1 - 4.59T + 11T^{2} \) |
| 13 | \( 1 - 2.27T + 13T^{2} \) |
| 17 | \( 1 + 7.35T + 17T^{2} \) |
| 19 | \( 1 + 1.09T + 19T^{2} \) |
| 23 | \( 1 + 1.49T + 23T^{2} \) |
| 29 | \( 1 - 8.48T + 29T^{2} \) |
| 31 | \( 1 + 5.03T + 31T^{2} \) |
| 37 | \( 1 - 6.78T + 37T^{2} \) |
| 41 | \( 1 - 8.03T + 41T^{2} \) |
| 43 | \( 1 + 4.20T + 43T^{2} \) |
| 47 | \( 1 - 4.34T + 47T^{2} \) |
| 53 | \( 1 + 6.51T + 53T^{2} \) |
| 59 | \( 1 + 1.11T + 59T^{2} \) |
| 61 | \( 1 + 5.92T + 61T^{2} \) |
| 67 | \( 1 + 8.49T + 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 - 16.0T + 73T^{2} \) |
| 79 | \( 1 + 1.02T + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 + 4.23T + 97T^{2} \) |
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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
−8.023181502676020295143994579709, −7.05971130824131524546986056295, −6.39554240352808957270508350190, −6.07155506590027869042064277833, −4.83569041107710197865799922138, −4.32044830428978305373201823855, −3.75712081071469970645813743974, −2.68041231711283820584627763741, −1.27382203151676590334174522376, 0,
1.27382203151676590334174522376, 2.68041231711283820584627763741, 3.75712081071469970645813743974, 4.32044830428978305373201823855, 4.83569041107710197865799922138, 6.07155506590027869042064277833, 6.39554240352808957270508350190, 7.05971130824131524546986056295, 8.023181502676020295143994579709
