| L(s) = 1 | + 4.07e3·2-s − 5.61e5·3-s + 8.23e6·4-s − 1.31e8·5-s − 2.29e9·6-s − 6.16e8·8-s + 2.21e11·9-s − 5.37e11·10-s − 1.04e12·11-s − 4.62e12·12-s − 1.36e12·13-s + 7.40e13·15-s − 7.16e13·16-s + 1.11e14·17-s + 9.03e14·18-s + 6.40e14·19-s − 1.08e15·20-s − 4.24e15·22-s + 7.20e14·23-s + 3.46e14·24-s + 5.43e15·25-s − 5.57e15·26-s − 7.15e16·27-s + 1.24e17·29-s + 3.01e17·30-s − 6.10e16·31-s − 2.86e17·32-s + ⋯ |
| L(s) = 1 | + 1.40·2-s − 1.83·3-s + 0.981·4-s − 1.20·5-s − 2.57·6-s − 0.0253·8-s + 2.35·9-s − 1.69·10-s − 1.10·11-s − 1.79·12-s − 0.211·13-s + 2.20·15-s − 1.01·16-s + 0.789·17-s + 3.31·18-s + 1.26·19-s − 1.18·20-s − 1.55·22-s + 0.157·23-s + 0.0464·24-s + 0.455·25-s − 0.297·26-s − 2.47·27-s + 1.89·29-s + 3.11·30-s − 0.431·31-s − 1.40·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 - 4.07e3T + 8.38e6T^{2} \) |
| 3 | \( 1 + 5.61e5T + 9.41e10T^{2} \) |
| 5 | \( 1 + 1.31e8T + 1.19e16T^{2} \) |
| 11 | \( 1 + 1.04e12T + 8.95e23T^{2} \) |
| 13 | \( 1 + 1.36e12T + 4.17e25T^{2} \) |
| 17 | \( 1 - 1.11e14T + 1.99e28T^{2} \) |
| 19 | \( 1 - 6.40e14T + 2.57e29T^{2} \) |
| 23 | \( 1 - 7.20e14T + 2.08e31T^{2} \) |
| 29 | \( 1 - 1.24e17T + 4.31e33T^{2} \) |
| 31 | \( 1 + 6.10e16T + 2.00e34T^{2} \) |
| 37 | \( 1 - 1.90e18T + 1.17e36T^{2} \) |
| 41 | \( 1 + 4.99e17T + 1.24e37T^{2} \) |
| 43 | \( 1 + 1.12e19T + 3.71e37T^{2} \) |
| 47 | \( 1 - 1.75e19T + 2.87e38T^{2} \) |
| 53 | \( 1 - 4.73e19T + 4.55e39T^{2} \) |
| 59 | \( 1 + 3.83e18T + 5.36e40T^{2} \) |
| 61 | \( 1 + 2.49e19T + 1.15e41T^{2} \) |
| 67 | \( 1 - 2.80e20T + 9.99e41T^{2} \) |
| 71 | \( 1 + 5.49e20T + 3.79e42T^{2} \) |
| 73 | \( 1 - 7.98e20T + 7.18e42T^{2} \) |
| 79 | \( 1 + 5.66e21T + 4.42e43T^{2} \) |
| 83 | \( 1 - 6.39e21T + 1.37e44T^{2} \) |
| 89 | \( 1 - 7.60e21T + 6.85e44T^{2} \) |
| 97 | \( 1 + 1.25e23T + 4.96e45T^{2} \) |
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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
−11.20908763173172077377287100372, −10.06180199633983240623218004545, −7.79849279729805643000492935892, −6.78358565867047613171578406496, −5.60733171700827440578692742925, −4.98733072404804690598605703692, −4.13116903560679982703713266952, −2.94342401031300831980660608623, −0.927514200546319723410393653376, 0,
0.927514200546319723410393653376, 2.94342401031300831980660608623, 4.13116903560679982703713266952, 4.98733072404804690598605703692, 5.60733171700827440578692742925, 6.78358565867047613171578406496, 7.79849279729805643000492935892, 10.06180199633983240623218004545, 11.20908763173172077377287100372
