| L(s) = 1 | − 0.595·2-s − 127.·4-s + 80.2·5-s − 870.·7-s + 152.·8-s − 47.7·10-s + 7.23e3·11-s − 1.16e4·13-s + 518.·14-s + 1.62e4·16-s + 1.97e4·17-s + 1.60e4·19-s − 1.02e4·20-s − 4.30e3·22-s + 1.21e4·23-s − 7.16e4·25-s + 6.94e3·26-s + 1.11e5·28-s + 2.13e5·29-s − 9.57e4·31-s − 2.91e4·32-s − 1.17e4·34-s − 6.98e4·35-s − 1.00e5·37-s − 9.53e3·38-s + 1.22e4·40-s − 3.53e5·41-s + ⋯ |
| L(s) = 1 | − 0.0526·2-s − 0.997·4-s + 0.287·5-s − 0.959·7-s + 0.105·8-s − 0.0151·10-s + 1.63·11-s − 1.47·13-s + 0.0504·14-s + 0.991·16-s + 0.974·17-s + 0.535·19-s − 0.286·20-s − 0.0862·22-s + 0.208·23-s − 0.917·25-s + 0.0774·26-s + 0.956·28-s + 1.62·29-s − 0.577·31-s − 0.157·32-s − 0.0512·34-s − 0.275·35-s − 0.325·37-s − 0.0281·38-s + 0.0301·40-s − 0.801·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 - 1.21e4T \) |
| good | 2 | \( 1 + 0.595T + 128T^{2} \) |
| 5 | \( 1 - 80.2T + 7.81e4T^{2} \) |
| 7 | \( 1 + 870.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 7.23e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.16e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.97e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.60e4T + 8.93e8T^{2} \) |
| 29 | \( 1 - 2.13e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 9.57e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.00e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.53e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.41e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 3.49e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.05e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.31e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.32e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 5.96e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.73e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.58e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.82e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.89e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 4.39e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 8.32e6T + 8.07e13T^{2} \) |
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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
−10.18953350067912026211768586226, −9.626318352637409155704790100133, −8.954998657519792561768778105613, −7.59347090005234094278317785071, −6.47534784079488220079475706673, −5.33420240058860985504835732906, −4.14703906373845772902818948978, −3.07183880963190292497142232796, −1.26672469051137571702088485199, 0,
1.26672469051137571702088485199, 3.07183880963190292497142232796, 4.14703906373845772902818948978, 5.33420240058860985504835732906, 6.47534784079488220079475706673, 7.59347090005234094278317785071, 8.954998657519792561768778105613, 9.626318352637409155704790100133, 10.18953350067912026211768586226
