| L(s) = 1 | − 17.0·3-s − 74.6·5-s − 68.6·7-s + 46.5·9-s − 159.·11-s + 193.·13-s + 1.27e3·15-s + 190.·17-s + 1.16e3·19-s + 1.16e3·21-s − 529·23-s + 2.44e3·25-s + 3.34e3·27-s + 3.72e3·29-s + 1.34e3·31-s + 2.70e3·33-s + 5.12e3·35-s − 1.16e3·37-s − 3.30e3·39-s − 1.49e4·41-s + 1.42e4·43-s − 3.47e3·45-s + 1.00e4·47-s − 1.20e4·49-s − 3.24e3·51-s − 2.57e4·53-s + 1.18e4·55-s + ⋯ |
| L(s) = 1 | − 1.09·3-s − 1.33·5-s − 0.529·7-s + 0.191·9-s − 0.396·11-s + 0.318·13-s + 1.45·15-s + 0.160·17-s + 0.737·19-s + 0.578·21-s − 0.208·23-s + 0.783·25-s + 0.882·27-s + 0.822·29-s + 0.251·31-s + 0.432·33-s + 0.707·35-s − 0.140·37-s − 0.347·39-s − 1.38·41-s + 1.17·43-s − 0.255·45-s + 0.664·47-s − 0.719·49-s − 0.174·51-s − 1.26·53-s + 0.529·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5831655586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5831655586\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + 529T \) |
| good | 3 | \( 1 + 17.0T + 243T^{2} \) |
| 5 | \( 1 + 74.6T + 3.12e3T^{2} \) |
| 7 | \( 1 + 68.6T + 1.68e4T^{2} \) |
| 11 | \( 1 + 159.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 193.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 190.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.16e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 3.72e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.34e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.16e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.49e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.42e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.00e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.57e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 967.T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.06e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.60e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.88e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.68e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.01e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.38e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.40e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.72e4T + 8.58e9T^{2} \) |
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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
−12.73490011167862601418878487140, −11.88677630204669728716689607293, −11.20461227232081993215885560704, −10.10682360348157034319629008408, −8.493229421199725669847406581114, −7.32093670214553163835965337794, −6.08762656396737846101137024574, −4.76995473407866603360595538840, −3.31287619356490240476198455428, −0.57517370250827340953864886244,
0.57517370250827340953864886244, 3.31287619356490240476198455428, 4.76995473407866603360595538840, 6.08762656396737846101137024574, 7.32093670214553163835965337794, 8.493229421199725669847406581114, 10.10682360348157034319629008408, 11.20461227232081993215885560704, 11.88677630204669728716689607293, 12.73490011167862601418878487140
