| L(s) = 1 | − 4.52·2-s − 1.75·3-s + 12.4·4-s − 13.0·5-s + 7.91·6-s + 4.32·7-s − 20.0·8-s − 23.9·9-s + 58.9·10-s − 55.2·11-s − 21.7·12-s − 37.7·13-s − 19.5·14-s + 22.8·15-s − 8.85·16-s − 24.9·17-s + 108.·18-s + 39.7·19-s − 162.·20-s − 7.57·21-s + 249.·22-s − 2.55·23-s + 35.1·24-s + 45.2·25-s + 170.·26-s + 89.1·27-s + 53.8·28-s + ⋯ |
| L(s) = 1 | − 1.59·2-s − 0.337·3-s + 1.55·4-s − 1.16·5-s + 0.538·6-s + 0.233·7-s − 0.886·8-s − 0.886·9-s + 1.86·10-s − 1.51·11-s − 0.523·12-s − 0.805·13-s − 0.373·14-s + 0.393·15-s − 0.138·16-s − 0.355·17-s + 1.41·18-s + 0.479·19-s − 1.81·20-s − 0.0787·21-s + 2.42·22-s − 0.0231·23-s + 0.298·24-s + 0.361·25-s + 1.28·26-s + 0.635·27-s + 0.363·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 + 4.52T + 8T^{2} \) |
| 3 | \( 1 + 1.75T + 27T^{2} \) |
| 5 | \( 1 + 13.0T + 125T^{2} \) |
| 7 | \( 1 - 4.32T + 343T^{2} \) |
| 11 | \( 1 + 55.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 37.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 24.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 39.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 2.55T + 1.21e4T^{2} \) |
| 29 | \( 1 + 238.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 263.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 353.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 254.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 386.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 35.7T + 1.48e5T^{2} \) |
| 59 | \( 1 + 87.4T + 2.05e5T^{2} \) |
| 61 | \( 1 - 890.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 998.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 443.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 70.5T + 3.89e5T^{2} \) |
| 79 | \( 1 - 975.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 836.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.08e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.13e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.247518752959554632551399326740, −7.903247342097837171953951864223, −7.43301394083697658747133741864, −6.38101686693347130259747263325, −5.29647442526563424100151693763, −4.46110722593405980695519180638, −3.02886040395183892884935578404, −2.23630854055838839734230141502, −0.67266851786011116871264386713, 0,
0.67266851786011116871264386713, 2.23630854055838839734230141502, 3.02886040395183892884935578404, 4.46110722593405980695519180638, 5.29647442526563424100151693763, 6.38101686693347130259747263325, 7.43301394083697658747133741864, 7.903247342097837171953951864223, 8.247518752959554632551399326740
