| L(s) = 1 | + 8·2-s + 40.5·3-s + 64·4-s − 147.·5-s + 324.·6-s − 419.·7-s + 512·8-s − 541.·9-s − 1.17e3·10-s + 240.·11-s + 2.59e3·12-s + 4.97e3·13-s − 3.35e3·14-s − 5.97e3·15-s + 4.09e3·16-s + 3.86e4·17-s − 4.32e3·18-s − 2.03e4·19-s − 9.42e3·20-s − 1.70e4·21-s + 1.92e3·22-s − 9.31e4·23-s + 2.07e4·24-s − 5.64e4·25-s + 3.98e4·26-s − 1.10e5·27-s − 2.68e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.867·3-s + 0.5·4-s − 0.527·5-s + 0.613·6-s − 0.462·7-s + 0.353·8-s − 0.247·9-s − 0.372·10-s + 0.0543·11-s + 0.433·12-s + 0.628·13-s − 0.327·14-s − 0.457·15-s + 0.250·16-s + 1.91·17-s − 0.174·18-s − 0.680·19-s − 0.263·20-s − 0.401·21-s + 0.0384·22-s − 1.59·23-s + 0.306·24-s − 0.722·25-s + 0.444·26-s − 1.08·27-s − 0.231·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8T \) |
| 269 | \( 1 + 1.94e7T \) |
| good | 3 | \( 1 - 40.5T + 2.18e3T^{2} \) |
| 5 | \( 1 + 147.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 419.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 240.T + 1.94e7T^{2} \) |
| 13 | \( 1 - 4.97e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.86e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.03e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 9.31e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 5.58e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.76e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.68e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 7.77e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.67e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 9.50e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 6.47e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.62e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.11e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.16e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.64e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.73e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.63e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 7.23e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.29e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 6.86e6T + 8.07e13T^{2} \) |
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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
−9.225549655296827597592928801748, −8.018085208648704216650687928610, −7.77409516589317450073821041055, −6.31144949911515690708936260129, −5.65761010172984738056306951898, −4.19799477328028941126789698143, −3.52004418387981647549755471913, −2.73777898215251919490825151672, −1.51074292337626808425169444711, 0,
1.51074292337626808425169444711, 2.73777898215251919490825151672, 3.52004418387981647549755471913, 4.19799477328028941126789698143, 5.65761010172984738056306951898, 6.31144949911515690708936260129, 7.77409516589317450073821041055, 8.018085208648704216650687928610, 9.225549655296827597592928801748
