| L(s) = 1 | − 10.3·2-s − 9·3-s + 75.4·4-s − 24.0·5-s + 93.2·6-s − 107.·7-s − 450.·8-s + 81·9-s + 249.·10-s − 183.·11-s − 678.·12-s + 1.15e3·13-s + 1.11e3·14-s + 216.·15-s + 2.25e3·16-s − 886.·17-s − 839.·18-s − 2.10e3·19-s − 1.81e3·20-s + 968.·21-s + 1.90e3·22-s − 1.56e3·23-s + 4.05e3·24-s − 2.54e3·25-s − 1.19e4·26-s − 729·27-s − 8.11e3·28-s + ⋯ |
| L(s) = 1 | − 1.83·2-s − 0.577·3-s + 2.35·4-s − 0.429·5-s + 1.05·6-s − 0.830·7-s − 2.48·8-s + 0.333·9-s + 0.787·10-s − 0.457·11-s − 1.36·12-s + 1.89·13-s + 1.52·14-s + 0.248·15-s + 2.20·16-s − 0.743·17-s − 0.610·18-s − 1.33·19-s − 1.01·20-s + 0.479·21-s + 0.837·22-s − 0.618·23-s + 1.43·24-s − 0.815·25-s − 3.47·26-s − 0.192·27-s − 1.95·28-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
| \(L(3)\) |
\(\approx\) |
\(0.2823258792\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2823258792\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 10.3T + 32T^{2} \) |
| 5 | \( 1 + 24.0T + 3.12e3T^{2} \) |
| 7 | \( 1 + 107.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 183.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.15e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 886.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.10e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.56e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.23e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.50e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.16e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.59e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.23e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.63e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.21e4T + 4.18e8T^{2} \) |
| 61 | \( 1 + 7.39e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.90e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.29e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.85e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.78e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.21e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.30e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.20e4T + 8.58e9T^{2} \) |
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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
−11.19107581599416782239102883113, −10.81084671538037879705174761830, −9.766286722459138756012685844835, −8.752101674270351480115295517803, −7.959132096596310287972215482418, −6.69540517380480932672404369829, −6.03673569874933005757267485338, −3.74402675058349569590745696322, −1.95843377023923889517315112176, −0.43853937049258350782184085579,
0.43853937049258350782184085579, 1.95843377023923889517315112176, 3.74402675058349569590745696322, 6.03673569874933005757267485338, 6.69540517380480932672404369829, 7.959132096596310287972215482418, 8.752101674270351480115295517803, 9.766286722459138756012685844835, 10.81084671538037879705174761830, 11.19107581599416782239102883113
