| L(s) = 1 | − 1.00e4·2-s + 7.41e5·3-s + 6.82e7·4-s + 7.48e8·5-s − 7.48e9·6-s − 3.50e11·8-s − 2.97e11·9-s − 7.55e12·10-s + 1.30e13·11-s + 5.06e13·12-s + 1.14e14·13-s + 5.55e14·15-s + 1.24e15·16-s + 8.99e14·17-s + 3.00e15·18-s − 1.76e16·19-s + 5.11e16·20-s − 1.31e17·22-s + 1.17e16·23-s − 2.59e17·24-s + 2.62e17·25-s − 1.15e18·26-s − 8.48e17·27-s − 2.64e18·29-s − 5.60e18·30-s + 2.61e18·31-s − 8.20e17·32-s + ⋯ |
| L(s) = 1 | − 1.74·2-s + 0.805·3-s + 2.03·4-s + 1.37·5-s − 1.40·6-s − 1.80·8-s − 0.351·9-s − 2.38·10-s + 1.25·11-s + 1.63·12-s + 1.36·13-s + 1.10·15-s + 1.10·16-s + 0.374·17-s + 0.612·18-s − 1.82·19-s + 2.79·20-s − 2.18·22-s + 0.111·23-s − 1.45·24-s + 0.881·25-s − 2.37·26-s − 1.08·27-s − 1.38·29-s − 1.92·30-s + 0.596·31-s − 0.125·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(2.013814975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.013814975\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + 1.00e4T + 3.35e7T^{2} \) |
| 3 | \( 1 - 7.41e5T + 8.47e11T^{2} \) |
| 5 | \( 1 - 7.48e8T + 2.98e17T^{2} \) |
| 11 | \( 1 - 1.30e13T + 1.08e26T^{2} \) |
| 13 | \( 1 - 1.14e14T + 7.05e27T^{2} \) |
| 17 | \( 1 - 8.99e14T + 5.77e30T^{2} \) |
| 19 | \( 1 + 1.76e16T + 9.30e31T^{2} \) |
| 23 | \( 1 - 1.17e16T + 1.10e34T^{2} \) |
| 29 | \( 1 + 2.64e18T + 3.63e36T^{2} \) |
| 31 | \( 1 - 2.61e18T + 1.92e37T^{2} \) |
| 37 | \( 1 - 1.53e19T + 1.60e39T^{2} \) |
| 41 | \( 1 + 7.08e19T + 2.08e40T^{2} \) |
| 43 | \( 1 - 3.68e20T + 6.86e40T^{2} \) |
| 47 | \( 1 + 1.15e21T + 6.34e41T^{2} \) |
| 53 | \( 1 - 7.12e21T + 1.27e43T^{2} \) |
| 59 | \( 1 + 5.10e21T + 1.86e44T^{2} \) |
| 61 | \( 1 - 6.66e21T + 4.29e44T^{2} \) |
| 67 | \( 1 - 9.02e21T + 4.48e45T^{2} \) |
| 71 | \( 1 - 1.81e23T + 1.91e46T^{2} \) |
| 73 | \( 1 - 3.01e23T + 3.82e46T^{2} \) |
| 79 | \( 1 + 3.98e23T + 2.75e47T^{2} \) |
| 83 | \( 1 - 1.81e24T + 9.48e47T^{2} \) |
| 89 | \( 1 - 2.96e23T + 5.42e48T^{2} \) |
| 97 | \( 1 - 6.37e24T + 4.66e49T^{2} \) |
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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
−10.49708449108225905763526791500, −9.404616236710335838208444894193, −8.919297617230673701396261432806, −8.120849394912336064454872903549, −6.61341638009415632486342909633, −5.95437140734702240394507326737, −3.70918259882122729210185171800, −2.29933586161611985967627526516, −1.74291549772279875746793006110, −0.76696753941859312778331653154,
0.76696753941859312778331653154, 1.74291549772279875746793006110, 2.29933586161611985967627526516, 3.70918259882122729210185171800, 5.95437140734702240394507326737, 6.61341638009415632486342909633, 8.120849394912336064454872903549, 8.919297617230673701396261432806, 9.404616236710335838208444894193, 10.49708449108225905763526791500
