| L(s) = 1 | − 8·2-s − 27·3-s + 64·4-s + 335.·5-s + 216·6-s − 140.·7-s − 512·8-s + 729·9-s − 2.68e3·10-s + 1.82e3·11-s − 1.72e3·12-s − 5.89e3·13-s + 1.12e3·14-s − 9.06e3·15-s + 4.09e3·16-s + 3.06e4·17-s − 5.83e3·18-s + 1.37e4·19-s + 2.14e4·20-s + 3.79e3·21-s − 1.45e4·22-s − 5.70e4·23-s + 1.38e4·24-s + 3.44e4·25-s + 4.71e4·26-s − 1.96e4·27-s − 9.00e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.20·5-s + 0.408·6-s − 0.155·7-s − 0.353·8-s + 0.333·9-s − 0.848·10-s + 0.413·11-s − 0.288·12-s − 0.744·13-s + 0.109·14-s − 0.693·15-s + 0.250·16-s + 1.51·17-s − 0.235·18-s + 0.458·19-s + 0.600·20-s + 0.0895·21-s − 0.292·22-s − 0.977·23-s + 0.204·24-s + 0.441·25-s + 0.526·26-s − 0.192·27-s − 0.0775·28-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 | \( 1 - 335.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 140.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.82e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 5.89e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.06e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.37e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 5.70e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.96e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.71e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.65e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 5.14e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.45e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 2.47e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.62e6T + 1.17e12T^{2} \) |
| 61 | \( 1 - 7.78e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.07e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 9.56e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.12e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.12e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.52e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 3.07e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 8.41e5T + 8.07e13T^{2} \) |
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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
−9.740911425983050922209496746095, −9.293672739232818623824336936375, −7.84562044624632021104771821412, −6.99604448999976581396256257496, −5.85652522204647646437099562256, −5.36930051064612802524791396971, −3.64120183653185580303680390932, −2.17500613019882052420010025474, −1.30187308388700230242192108470, 0,
1.30187308388700230242192108470, 2.17500613019882052420010025474, 3.64120183653185580303680390932, 5.36930051064612802524791396971, 5.85652522204647646437099562256, 6.99604448999976581396256257496, 7.84562044624632021104771821412, 9.293672739232818623824336936375, 9.740911425983050922209496746095
