| L(s) = 1 | − 2.43·3-s − 13.0·5-s + 7·7-s − 21.0·9-s − 1.69·11-s − 9.51·13-s + 31.6·15-s + 44.0·17-s + 84.5·19-s − 17.0·21-s + 86.7·23-s + 44.3·25-s + 117.·27-s + 185.·29-s − 21.8·31-s + 4.12·33-s − 91.0·35-s − 341.·37-s + 23.1·39-s + 41·41-s + 296.·43-s + 274.·45-s − 305.·47-s + 49·49-s − 107.·51-s − 634.·53-s + 22.0·55-s + ⋯ |
| L(s) = 1 | − 0.468·3-s − 1.16·5-s + 0.377·7-s − 0.780·9-s − 0.0464·11-s − 0.203·13-s + 0.545·15-s + 0.628·17-s + 1.02·19-s − 0.177·21-s + 0.786·23-s + 0.354·25-s + 0.834·27-s + 1.18·29-s − 0.126·31-s + 0.0217·33-s − 0.439·35-s − 1.51·37-s + 0.0951·39-s + 0.156·41-s + 1.05·43-s + 0.908·45-s − 0.946·47-s + 0.142·49-s − 0.294·51-s − 1.64·53-s + 0.0540·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 41 | \( 1 - 41T \) |
| good | 3 | \( 1 + 2.43T + 27T^{2} \) |
| 5 | \( 1 + 13.0T + 125T^{2} \) |
| 11 | \( 1 + 1.69T + 1.33e3T^{2} \) |
| 13 | \( 1 + 9.51T + 2.19e3T^{2} \) |
| 17 | \( 1 - 44.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 84.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 86.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 185.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 21.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 341.T + 5.06e4T^{2} \) |
| 43 | \( 1 - 296.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 305.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 634.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 604.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 247.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 723.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 172.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 718.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 866.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 808.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 781.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 680.T + 9.12e5T^{2} \) |
| 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
−8.860390634525833849800586605996, −8.120448163320519064406919111693, −7.46451837682571797032284603718, −6.56094370532607794265318204097, −5.42452005090019092888713307111, −4.82832980985812278932750529773, −3.65794216177514197958812676803, −2.81950584364448941813340125907, −1.11796064208652800365252818499, 0,
1.11796064208652800365252818499, 2.81950584364448941813340125907, 3.65794216177514197958812676803, 4.82832980985812278932750529773, 5.42452005090019092888713307111, 6.56094370532607794265318204097, 7.46451837682571797032284603718, 8.120448163320519064406919111693, 8.860390634525833849800586605996
