L(s) = 1 | − 2-s − 3-s + 4-s − 2·5-s + 6-s − 4·7-s − 8-s + 9-s + 2·10-s + 11-s − 12-s + 6·13-s + 4·14-s + 2·15-s + 16-s − 2·17-s − 18-s − 2·20-s + 4·21-s − 22-s − 8·23-s + 24-s − 25-s − 6·26-s − 27-s − 4·28-s + 10·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.301·11-s − 0.288·12-s + 1.66·13-s + 1.06·14-s + 0.516·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.447·20-s + 0.872·21-s − 0.213·22-s − 1.66·23-s + 0.204·24-s − 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.755·28-s + 1.85·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2442 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2442 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 37 | \( 1 - T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.391033343949074591800273823157, −8.058130128466856535670508644110, −6.79157445288647411476511276058, −6.43706535819598929287205410560, −5.82846188728294037504645538815, −4.29390675380669768429331154375, −3.71864997468122498159094755631, −2.70993877353183061510862664825, −1.10928922969834662863813252585, 0,
1.10928922969834662863813252585, 2.70993877353183061510862664825, 3.71864997468122498159094755631, 4.29390675380669768429331154375, 5.82846188728294037504645538815, 6.43706535819598929287205410560, 6.79157445288647411476511276058, 8.058130128466856535670508644110, 8.391033343949074591800273823157