| L(s) = 1 | + 7.32·2-s − 9·3-s + 21.6·4-s + 46.7·5-s − 65.9·6-s − 85.8·7-s − 75.6·8-s + 81·9-s + 342.·10-s − 408.·11-s − 195.·12-s + 27.9·13-s − 628.·14-s − 421.·15-s − 1.24e3·16-s + 1.58e3·17-s + 593.·18-s − 2.20e3·19-s + 1.01e3·20-s + 772.·21-s − 2.99e3·22-s − 3.96e3·23-s + 681.·24-s − 936.·25-s + 204.·26-s − 729·27-s − 1.85e3·28-s + ⋯ |
| L(s) = 1 | + 1.29·2-s − 0.577·3-s + 0.677·4-s + 0.836·5-s − 0.747·6-s − 0.661·7-s − 0.418·8-s + 0.333·9-s + 1.08·10-s − 1.01·11-s − 0.390·12-s + 0.0458·13-s − 0.857·14-s − 0.483·15-s − 1.21·16-s + 1.33·17-s + 0.431·18-s − 1.40·19-s + 0.566·20-s + 0.382·21-s − 1.31·22-s − 1.56·23-s + 0.241·24-s − 0.299·25-s + 0.0593·26-s − 0.192·27-s − 0.448·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 59 | \( 1 + 3.48e3T \) |
| good | 2 | \( 1 - 7.32T + 32T^{2} \) |
| 5 | \( 1 - 46.7T + 3.12e3T^{2} \) |
| 7 | \( 1 + 85.8T + 1.68e4T^{2} \) |
| 11 | \( 1 + 408.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 27.9T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.58e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.20e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.96e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.90e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.85e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.83e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.69e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.79e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.10e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.10e4T + 4.18e8T^{2} \) |
| 61 | \( 1 - 4.53e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.08e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.70e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.04e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.16e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.32e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.16e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.11e4T + 8.58e9T^{2} \) |
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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
−11.64364431164781184450173082456, −10.32884287222278158240977178636, −9.658397715672460033710692055148, −8.032756850079619439179142868150, −6.39218593494055897448469229840, −5.86991269398637464229964897234, −4.88020458898809367880637529565, −3.56553671675757864951153626798, −2.19858337587210411237660967552, 0,
2.19858337587210411237660967552, 3.56553671675757864951153626798, 4.88020458898809367880637529565, 5.86991269398637464229964897234, 6.39218593494055897448469229840, 8.032756850079619439179142868150, 9.658397715672460033710692055148, 10.32884287222278158240977178636, 11.64364431164781184450173082456
