| L(s) = 1 | − 2.02e3·2-s − 2.32e5·3-s − 4.27e6·4-s + 9.33e7·5-s + 4.72e8·6-s + 2.56e10·8-s − 3.98e10·9-s − 1.89e11·10-s + 9.02e11·11-s + 9.94e11·12-s + 2.46e12·13-s − 2.17e13·15-s − 1.62e13·16-s − 1.21e14·17-s + 8.09e13·18-s − 1.65e14·19-s − 3.98e14·20-s − 1.83e15·22-s + 6.91e15·23-s − 5.98e15·24-s − 3.19e15·25-s − 5.00e15·26-s + 3.12e16·27-s − 1.21e17·29-s + 4.41e16·30-s − 1.02e17·31-s − 1.82e17·32-s + ⋯ |
| L(s) = 1 | − 0.700·2-s − 0.759·3-s − 0.509·4-s + 0.855·5-s + 0.531·6-s + 1.05·8-s − 0.423·9-s − 0.599·10-s + 0.953·11-s + 0.386·12-s + 0.381·13-s − 0.649·15-s − 0.231·16-s − 0.859·17-s + 0.296·18-s − 0.325·19-s − 0.435·20-s − 0.668·22-s + 1.51·23-s − 0.802·24-s − 0.268·25-s − 0.267·26-s + 1.08·27-s − 1.84·29-s + 0.454·30-s − 0.723·31-s − 0.895·32-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 | \( 1 + 2.02e3T + 8.38e6T^{2} \) |
| 3 | \( 1 + 2.32e5T + 9.41e10T^{2} \) |
| 5 | \( 1 - 9.33e7T + 1.19e16T^{2} \) |
| 11 | \( 1 - 9.02e11T + 8.95e23T^{2} \) |
| 13 | \( 1 - 2.46e12T + 4.17e25T^{2} \) |
| 17 | \( 1 + 1.21e14T + 1.99e28T^{2} \) |
| 19 | \( 1 + 1.65e14T + 2.57e29T^{2} \) |
| 23 | \( 1 - 6.91e15T + 2.08e31T^{2} \) |
| 29 | \( 1 + 1.21e17T + 4.31e33T^{2} \) |
| 31 | \( 1 + 1.02e17T + 2.00e34T^{2} \) |
| 37 | \( 1 + 1.32e18T + 1.17e36T^{2} \) |
| 41 | \( 1 - 2.18e18T + 1.24e37T^{2} \) |
| 43 | \( 1 - 4.71e18T + 3.71e37T^{2} \) |
| 47 | \( 1 - 7.28e18T + 2.87e38T^{2} \) |
| 53 | \( 1 - 3.91e19T + 4.55e39T^{2} \) |
| 59 | \( 1 + 8.72e19T + 5.36e40T^{2} \) |
| 61 | \( 1 - 4.86e20T + 1.15e41T^{2} \) |
| 67 | \( 1 + 3.29e20T + 9.99e41T^{2} \) |
| 71 | \( 1 - 3.32e20T + 3.79e42T^{2} \) |
| 73 | \( 1 + 3.57e21T + 7.18e42T^{2} \) |
| 79 | \( 1 - 5.37e21T + 4.42e43T^{2} \) |
| 83 | \( 1 - 1.47e22T + 1.37e44T^{2} \) |
| 89 | \( 1 - 2.01e22T + 6.85e44T^{2} \) |
| 97 | \( 1 + 1.18e23T + 4.96e45T^{2} \) |
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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
−10.60284573923746950192194448409, −9.268964971225296384497666820660, −8.818697681247069870726695730267, −7.17212626330202188492222124714, −6.01313729396035118841732136216, −5.09196651178113092399759226051, −3.80581035659919516449730366245, −2.03418492240265721680507598112, −1.00867934974759210100793742833, 0,
1.00867934974759210100793742833, 2.03418492240265721680507598112, 3.80581035659919516449730366245, 5.09196651178113092399759226051, 6.01313729396035118841732136216, 7.17212626330202188492222124714, 8.818697681247069870726695730267, 9.268964971225296384497666820660, 10.60284573923746950192194448409
