| L(s) = 1 | − 8·2-s + 64.7·3-s + 64·4-s − 436.·5-s − 517.·6-s − 168.·7-s − 512·8-s + 2.00e3·9-s + 3.48e3·10-s − 5.90e3·11-s + 4.14e3·12-s + 4.98e3·13-s + 1.35e3·14-s − 2.82e4·15-s + 4.09e3·16-s + 1.38e4·17-s − 1.60e4·18-s + 3.54e4·19-s − 2.79e4·20-s − 1.09e4·21-s + 4.72e4·22-s + 794.·23-s − 3.31e4·24-s + 1.12e5·25-s − 3.98e4·26-s − 1.20e4·27-s − 1.08e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.38·3-s + 0.5·4-s − 1.56·5-s − 0.978·6-s − 0.186·7-s − 0.353·8-s + 0.915·9-s + 1.10·10-s − 1.33·11-s + 0.691·12-s + 0.628·13-s + 0.131·14-s − 2.15·15-s + 0.250·16-s + 0.685·17-s − 0.647·18-s + 1.18·19-s − 0.780·20-s − 0.257·21-s + 0.946·22-s + 0.0136·23-s − 0.489·24-s + 1.43·25-s − 0.444·26-s − 0.117·27-s − 0.0930·28-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 | \( 1 - 64.7T + 2.18e3T^{2} \) |
| 5 | \( 1 + 436.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 168.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 5.90e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 4.98e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.38e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.54e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 794.T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.34e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.83e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.90e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.96e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.52e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.24e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 3.51e4T + 1.17e12T^{2} \) |
| 59 | \( 1 + 7.19e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 4.50e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.63e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.89e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.66e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.95e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.27e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 6.84e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.07e6T + 8.07e13T^{2} \) |
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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
−9.018626829908871255952064118805, −8.208400702806391412426210870411, −7.76436441323046331639059024888, −7.19697706434239061326632652292, −5.57765320761655949895918387344, −4.12968609341996431393932687645, −3.24089932177548019645144658528, −2.63911284277382821389997664787, −1.11972073492575605683185063231, 0,
1.11972073492575605683185063231, 2.63911284277382821389997664787, 3.24089932177548019645144658528, 4.12968609341996431393932687645, 5.57765320761655949895918387344, 7.19697706434239061326632652292, 7.76436441323046331639059024888, 8.208400702806391412426210870411, 9.018626829908871255952064118805
