L(s) = 1 | + 8·2-s − 27·3-s + 64·4-s − 490.·5-s − 216·6-s + 975.·7-s + 512·8-s + 729·9-s − 3.92e3·10-s − 7.23e3·11-s − 1.72e3·12-s + 5.31e3·13-s + 7.80e3·14-s + 1.32e4·15-s + 4.09e3·16-s + 6.97e3·17-s + 5.83e3·18-s + 4.12e4·19-s − 3.14e4·20-s − 2.63e4·21-s − 5.79e4·22-s − 3.38e4·23-s − 1.38e4·24-s + 1.62e5·25-s + 4.25e4·26-s − 1.96e4·27-s + 6.24e4·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.75·5-s − 0.408·6-s + 1.07·7-s + 0.353·8-s + 0.333·9-s − 1.24·10-s − 1.63·11-s − 0.288·12-s + 0.671·13-s + 0.759·14-s + 1.01·15-s + 0.250·16-s + 0.344·17-s + 0.235·18-s + 1.37·19-s − 0.877·20-s − 0.620·21-s − 1.15·22-s − 0.580·23-s − 0.204·24-s + 2.08·25-s + 0.474·26-s − 0.192·27-s + 0.537·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 8T \) |
| 3 | \( 1 + 27T \) |
| 59 | \( 1 + 2.05e5T \) |
good | 5 | \( 1 + 490.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 975.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 7.23e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 5.31e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 6.97e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.12e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 3.38e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 6.54e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 5.89e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.71e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.31e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.69e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 8.40e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 4.30e5T + 1.17e12T^{2} \) |
| 61 | \( 1 + 2.69e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.59e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.78e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.90e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.73e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.24e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.74e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 7.05e6T + 8.07e13T^{2} \) |
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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
−10.32586433767314265366549219450, −8.412305402286243125043950988345, −7.80344205849290973206664605270, −7.12625085562337011309771268168, −5.56249279832283564346921587509, −4.88653917483137975560896353420, −3.96265535939508839229899682583, −2.90220618857152639681191828578, −1.21569883211088824723722201094, 0,
1.21569883211088824723722201094, 2.90220618857152639681191828578, 3.96265535939508839229899682583, 4.88653917483137975560896353420, 5.56249279832283564346921587509, 7.12625085562337011309771268168, 7.80344205849290973206664605270, 8.412305402286243125043950988345, 10.32586433767314265366549219450