| L(s) = 1 | − 257.·2-s + 6.56e3·3-s − 6.47e4·4-s − 1.68e6·6-s − 8.43e6·7-s + 5.04e7·8-s + 4.30e7·9-s − 1.27e9·11-s − 4.24e8·12-s − 2.16e9·13-s + 2.17e9·14-s − 4.50e9·16-s − 3.94e10·17-s − 1.10e10·18-s + 1.12e11·19-s − 5.53e10·21-s + 3.28e11·22-s − 5.33e11·23-s + 3.30e11·24-s + 5.58e11·26-s + 2.82e11·27-s + 5.46e11·28-s − 1.70e12·29-s − 1.16e11·31-s − 5.44e12·32-s − 8.37e12·33-s + 1.01e13·34-s + ⋯ |
| L(s) = 1 | − 0.711·2-s + 0.577·3-s − 0.493·4-s − 0.410·6-s − 0.553·7-s + 1.06·8-s + 0.333·9-s − 1.79·11-s − 0.285·12-s − 0.737·13-s + 0.393·14-s − 0.262·16-s − 1.37·17-s − 0.237·18-s + 1.51·19-s − 0.319·21-s + 1.27·22-s − 1.41·23-s + 0.613·24-s + 0.524·26-s + 0.192·27-s + 0.273·28-s − 0.632·29-s − 0.0244·31-s − 0.876·32-s − 1.03·33-s + 0.974·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(0.3858429411\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3858429411\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 6.56e3T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 257.T + 1.31e5T^{2} \) |
| 7 | \( 1 + 8.43e6T + 2.32e14T^{2} \) |
| 11 | \( 1 + 1.27e9T + 5.05e17T^{2} \) |
| 13 | \( 1 + 2.16e9T + 8.65e18T^{2} \) |
| 17 | \( 1 + 3.94e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 1.12e11T + 5.48e21T^{2} \) |
| 23 | \( 1 + 5.33e11T + 1.41e23T^{2} \) |
| 29 | \( 1 + 1.70e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + 1.16e11T + 2.25e25T^{2} \) |
| 37 | \( 1 + 3.69e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 6.01e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 3.10e13T + 5.87e27T^{2} \) |
| 47 | \( 1 - 2.34e14T + 2.66e28T^{2} \) |
| 53 | \( 1 + 5.42e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 1.85e14T + 1.27e30T^{2} \) |
| 61 | \( 1 + 1.71e15T + 2.24e30T^{2} \) |
| 67 | \( 1 - 6.75e12T + 1.10e31T^{2} \) |
| 71 | \( 1 - 9.21e13T + 2.96e31T^{2} \) |
| 73 | \( 1 + 1.14e16T + 4.74e31T^{2} \) |
| 79 | \( 1 - 8.98e15T + 1.81e32T^{2} \) |
| 83 | \( 1 + 4.25e15T + 4.21e32T^{2} \) |
| 89 | \( 1 + 3.54e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 3.08e16T + 5.95e33T^{2} \) |
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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
−10.76426114473312750397103427893, −9.897725101024874395589158564136, −9.119250537126258051661894398760, −7.965836563287463047032778685174, −7.29074086122433977660262783926, −5.48859378724785992656446333292, −4.36997376907068022087247114398, −2.97699136068393620573479294716, −1.89855003971937103992837075310, −0.29184656463571752395965434864,
0.29184656463571752395965434864, 1.89855003971937103992837075310, 2.97699136068393620573479294716, 4.36997376907068022087247114398, 5.48859378724785992656446333292, 7.29074086122433977660262783926, 7.965836563287463047032778685174, 9.119250537126258051661894398760, 9.897725101024874395589158564136, 10.76426114473312750397103427893
