| L(s) = 1 | − 2.38·2-s − 26.3·4-s + 81.0·5-s + 5.57·7-s + 139.·8-s − 193.·10-s − 121·11-s − 265.·13-s − 13.2·14-s + 510.·16-s + 1.71e3·17-s + 487.·19-s − 2.13e3·20-s + 288.·22-s − 4.67e3·23-s + 3.44e3·25-s + 632.·26-s − 146.·28-s − 7.83e3·29-s + 872.·31-s − 5.66e3·32-s − 4.08e3·34-s + 452.·35-s + 8.79e3·37-s − 1.16e3·38-s + 1.12e4·40-s + 1.68e4·41-s + ⋯ |
| L(s) = 1 | − 0.421·2-s − 0.822·4-s + 1.45·5-s + 0.0430·7-s + 0.767·8-s − 0.611·10-s − 0.301·11-s − 0.435·13-s − 0.0181·14-s + 0.498·16-s + 1.43·17-s + 0.309·19-s − 1.19·20-s + 0.127·22-s − 1.84·23-s + 1.10·25-s + 0.183·26-s − 0.0353·28-s − 1.72·29-s + 0.163·31-s − 0.978·32-s − 0.606·34-s + 0.0623·35-s + 1.05·37-s − 0.130·38-s + 1.11·40-s + 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 + 2.38T + 32T^{2} \) |
| 5 | \( 1 - 81.0T + 3.12e3T^{2} \) |
| 7 | \( 1 - 5.57T + 1.68e4T^{2} \) |
| 13 | \( 1 + 265.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.71e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 487.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.67e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.83e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 872.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.79e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.68e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.09e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.43e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.93e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.69e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.50e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.82e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.15e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.63e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.86e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.62e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.97e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.44e4T + 8.58e9T^{2} \) |
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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
−9.334404997381239318590892658510, −8.066295520474791364965923565352, −7.58548406415343141259251430986, −6.08834708770455847934782485225, −5.58149959478536568121094229711, −4.66574632196087718728208978262, −3.45997222153153123791955135688, −2.13847023552181554349795160824, −1.26807872309724389128691024355, 0,
1.26807872309724389128691024355, 2.13847023552181554349795160824, 3.45997222153153123791955135688, 4.66574632196087718728208978262, 5.58149959478536568121094229711, 6.08834708770455847934782485225, 7.58548406415343141259251430986, 8.066295520474791364965923565352, 9.334404997381239318590892658510
