| L(s) = 1 | − 8.53·2-s + 27·3-s − 55.0·4-s − 446.·5-s − 230.·6-s + 1.77e3·7-s + 1.56e3·8-s + 729·9-s + 3.81e3·10-s − 5.07e3·11-s − 1.48e3·12-s + 1.04e4·13-s − 1.51e4·14-s − 1.20e4·15-s − 6.30e3·16-s − 976.·17-s − 6.22e3·18-s − 4.79e4·19-s + 2.45e4·20-s + 4.77e4·21-s + 4.33e4·22-s − 1.21e4·23-s + 4.22e4·24-s + 1.20e5·25-s − 8.90e4·26-s + 1.96e4·27-s − 9.74e4·28-s + ⋯ |
| L(s) = 1 | − 0.754·2-s + 0.577·3-s − 0.430·4-s − 1.59·5-s − 0.435·6-s + 1.95·7-s + 1.07·8-s + 0.333·9-s + 1.20·10-s − 1.14·11-s − 0.248·12-s + 1.31·13-s − 1.47·14-s − 0.921·15-s − 0.384·16-s − 0.0482·17-s − 0.251·18-s − 1.60·19-s + 0.686·20-s + 1.12·21-s + 0.868·22-s − 0.208·23-s + 0.623·24-s + 1.54·25-s − 0.993·26-s + 0.192·27-s − 0.839·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 27T \) |
| 23 | \( 1 + 1.21e4T \) |
| good | 2 | \( 1 + 8.53T + 128T^{2} \) |
| 5 | \( 1 + 446.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.77e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 5.07e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.04e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 976.T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.79e4T + 8.93e8T^{2} \) |
| 29 | \( 1 + 7.07e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.07e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.00e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.51e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.61e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 4.14e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.16e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.73e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.22e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.16e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.37e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.18e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.55e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 4.67e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.99e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 2.86e6T + 8.07e13T^{2} \) |
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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
−12.73653751276774651178936347282, −11.17280067219446102653865723604, −10.70820435526574583228336995065, −8.644588458189265474107640883524, −8.234338816005188208876555180738, −7.55602024880720791938624531374, −4.85599278304017360325565667356, −3.90767091063866880363785371827, −1.62994871660570283132174531753, 0,
1.62994871660570283132174531753, 3.90767091063866880363785371827, 4.85599278304017360325565667356, 7.55602024880720791938624531374, 8.234338816005188208876555180738, 8.644588458189265474107640883524, 10.70820435526574583228336995065, 11.17280067219446102653865723604, 12.73653751276774651178936347282
