| L(s) = 1 | − 4.09e3·2-s − 1.75e6·3-s + 1.67e7·4-s + 7.20e9·6-s + 3.04e10·7-s − 6.87e10·8-s + 2.24e12·9-s + 2.58e12·11-s − 2.94e13·12-s + 9.57e13·13-s − 1.24e14·14-s + 2.81e14·16-s + 1.64e15·17-s − 9.18e15·18-s + 4.95e15·19-s − 5.34e16·21-s − 1.05e16·22-s + 1.07e16·23-s + 1.20e17·24-s − 3.92e17·26-s − 2.45e18·27-s + 5.10e17·28-s − 1.36e18·29-s − 4.41e18·31-s − 1.15e18·32-s − 4.54e18·33-s − 6.74e18·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.90·3-s + 0.5·4-s + 1.35·6-s + 0.830·7-s − 0.353·8-s + 2.64·9-s + 0.248·11-s − 0.954·12-s + 1.13·13-s − 0.587·14-s + 0.250·16-s + 0.685·17-s − 1.87·18-s + 0.513·19-s − 1.58·21-s − 0.175·22-s + 0.102·23-s + 0.675·24-s − 0.805·26-s − 3.14·27-s + 0.415·28-s − 0.717·29-s − 1.00·31-s − 0.176·32-s − 0.474·33-s − 0.484·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4.09e3T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 1.75e6T + 8.47e11T^{2} \) |
| 7 | \( 1 - 3.04e10T + 1.34e21T^{2} \) |
| 11 | \( 1 - 2.58e12T + 1.08e26T^{2} \) |
| 13 | \( 1 - 9.57e13T + 7.05e27T^{2} \) |
| 17 | \( 1 - 1.64e15T + 5.77e30T^{2} \) |
| 19 | \( 1 - 4.95e15T + 9.30e31T^{2} \) |
| 23 | \( 1 - 1.07e16T + 1.10e34T^{2} \) |
| 29 | \( 1 + 1.36e18T + 3.63e36T^{2} \) |
| 31 | \( 1 + 4.41e18T + 1.92e37T^{2} \) |
| 37 | \( 1 + 1.01e19T + 1.60e39T^{2} \) |
| 41 | \( 1 - 1.58e20T + 2.08e40T^{2} \) |
| 43 | \( 1 + 1.83e20T + 6.86e40T^{2} \) |
| 47 | \( 1 + 1.40e21T + 6.34e41T^{2} \) |
| 53 | \( 1 - 1.99e21T + 1.27e43T^{2} \) |
| 59 | \( 1 + 4.16e21T + 1.86e44T^{2} \) |
| 61 | \( 1 - 3.42e22T + 4.29e44T^{2} \) |
| 67 | \( 1 + 8.67e22T + 4.48e45T^{2} \) |
| 71 | \( 1 + 5.13e22T + 1.91e46T^{2} \) |
| 73 | \( 1 + 3.49e22T + 3.82e46T^{2} \) |
| 79 | \( 1 - 2.91e23T + 2.75e47T^{2} \) |
| 83 | \( 1 - 1.64e24T + 9.48e47T^{2} \) |
| 89 | \( 1 - 8.74e23T + 5.42e48T^{2} \) |
| 97 | \( 1 + 1.00e25T + 4.66e49T^{2} \) |
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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
−10.62551121825836687047960511456, −9.472231901410124564038491773144, −7.956256082449269039531899838696, −6.88846942323541698403999760851, −5.87675457435075388923794713264, −5.08725926764375289198714881171, −3.77060966191111058636399324325, −1.62884763983763429371875651334, −1.07294077704102346513981733557, 0,
1.07294077704102346513981733557, 1.62884763983763429371875651334, 3.77060966191111058636399324325, 5.08725926764375289198714881171, 5.87675457435075388923794713264, 6.88846942323541698403999760851, 7.956256082449269039531899838696, 9.472231901410124564038491773144, 10.62551121825836687047960511456
