| L(s) = 1 | − 10.9·2-s − 9·3-s + 86.8·4-s + 88.1·5-s + 98.1·6-s + 61.7·7-s − 598.·8-s + 81·9-s − 960.·10-s + 423.·11-s − 781.·12-s − 1.04e3·13-s − 672.·14-s − 793.·15-s + 3.74e3·16-s − 2.03e3·17-s − 883.·18-s − 1.53e3·19-s + 7.65e3·20-s − 555.·21-s − 4.61e3·22-s + 2.98e3·23-s + 5.38e3·24-s + 4.63e3·25-s + 1.14e4·26-s − 729·27-s + 5.36e3·28-s + ⋯ |
| L(s) = 1 | − 1.92·2-s − 0.577·3-s + 2.71·4-s + 1.57·5-s + 1.11·6-s + 0.476·7-s − 3.30·8-s + 0.333·9-s − 3.03·10-s + 1.05·11-s − 1.56·12-s − 1.71·13-s − 0.917·14-s − 0.910·15-s + 3.65·16-s − 1.70·17-s − 0.642·18-s − 0.973·19-s + 4.27·20-s − 0.274·21-s − 2.03·22-s + 1.17·23-s + 1.90·24-s + 1.48·25-s + 3.31·26-s − 0.192·27-s + 1.29·28-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 | \( 1 + 10.9T + 32T^{2} \) |
| 5 | \( 1 - 88.1T + 3.12e3T^{2} \) |
| 7 | \( 1 - 61.7T + 1.68e4T^{2} \) |
| 11 | \( 1 - 423.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.04e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.03e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.53e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.98e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 115.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.62e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.50e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 829.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.22e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.61e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.44e4T + 4.18e8T^{2} \) |
| 61 | \( 1 - 1.64e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 9.81e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.38e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.70e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.04e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.45e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.97e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.85e4T + 8.58e9T^{2} \) |
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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
−10.91195880880638164485399721657, −10.12218538800775077347514535740, −9.321356182542672203668874221968, −8.648770108381383298580749338193, −6.94645224988009614124550588068, −6.59756799380572094058236864619, −5.15690135088567038919501953935, −2.30381454022528759045812443170, −1.58418288688224332054055992137, 0,
1.58418288688224332054055992137, 2.30381454022528759045812443170, 5.15690135088567038919501953935, 6.59756799380572094058236864619, 6.94645224988009614124550588068, 8.648770108381383298580749338193, 9.321356182542672203668874221968, 10.12218538800775077347514535740, 10.91195880880638164485399721657
