| L(s) = 1 | − 0.0881·2-s + 3.20·3-s − 1.99·4-s − 0.282·6-s + 1.95·7-s + 0.351·8-s + 7.26·9-s − 11-s − 6.38·12-s − 3.20·13-s − 0.172·14-s + 3.95·16-s − 0.503·17-s − 0.640·18-s + 19-s + 6.25·21-s + 0.0881·22-s − 1.05·23-s + 1.12·24-s + 0.282·26-s + 13.6·27-s − 3.88·28-s − 7.91·29-s + 9.52·31-s − 1.05·32-s − 3.20·33-s + 0.0444·34-s + ⋯ |
| L(s) = 1 | − 0.0623·2-s + 1.84·3-s − 0.996·4-s − 0.115·6-s + 0.737·7-s + 0.124·8-s + 2.42·9-s − 0.301·11-s − 1.84·12-s − 0.889·13-s − 0.0459·14-s + 0.988·16-s − 0.122·17-s − 0.150·18-s + 0.229·19-s + 1.36·21-s + 0.0187·22-s − 0.220·23-s + 0.230·24-s + 0.0554·26-s + 2.62·27-s − 0.734·28-s − 1.46·29-s + 1.71·31-s − 0.186·32-s − 0.557·33-s + 0.00761·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.383795161\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.383795161\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 0.0881T + 2T^{2} \) |
| 3 | \( 1 - 3.20T + 3T^{2} \) |
| 7 | \( 1 - 1.95T + 7T^{2} \) |
| 13 | \( 1 + 3.20T + 13T^{2} \) |
| 17 | \( 1 + 0.503T + 17T^{2} \) |
| 23 | \( 1 + 1.05T + 23T^{2} \) |
| 29 | \( 1 + 7.91T + 29T^{2} \) |
| 31 | \( 1 - 9.52T + 31T^{2} \) |
| 37 | \( 1 - 5.22T + 37T^{2} \) |
| 41 | \( 1 - 8.32T + 41T^{2} \) |
| 43 | \( 1 - 8.04T + 43T^{2} \) |
| 47 | \( 1 - 9.39T + 47T^{2} \) |
| 53 | \( 1 + 2.37T + 53T^{2} \) |
| 59 | \( 1 + 2.80T + 59T^{2} \) |
| 61 | \( 1 + 1.53T + 61T^{2} \) |
| 67 | \( 1 - 9.79T + 67T^{2} \) |
| 71 | \( 1 + 4.45T + 71T^{2} \) |
| 73 | \( 1 - 1.55T + 73T^{2} \) |
| 79 | \( 1 - 3.00T + 79T^{2} \) |
| 83 | \( 1 - 2.22T + 83T^{2} \) |
| 89 | \( 1 - 6.10T + 89T^{2} \) |
| 97 | \( 1 - 4.69T + 97T^{2} \) |
| 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
−8.110619651858157412723337102639, −7.77734866256241517284942023535, −7.29683544537615590013359098252, −5.99129819944885544988406025884, −4.93447911365405427935608860989, −4.38978899331355348473531774541, −3.74734054949975880979652879261, −2.77061078574194524986136852395, −2.12475522881501442211525193575, −0.964258681392017245168158222203,
0.964258681392017245168158222203, 2.12475522881501442211525193575, 2.77061078574194524986136852395, 3.74734054949975880979652879261, 4.38978899331355348473531774541, 4.93447911365405427935608860989, 5.99129819944885544988406025884, 7.29683544537615590013359098252, 7.77734866256241517284942023535, 8.110619651858157412723337102639
