| L(s) = 1 | − 2·2-s + 4·4-s + 5·5-s + 8·7-s − 8·8-s − 10·10-s − 12·11-s + 13·13-s − 16·14-s + 16·16-s + 42·17-s − 52·19-s + 20·20-s + 24·22-s − 132·23-s + 25·25-s − 26·26-s + 32·28-s − 282·29-s + 116·31-s − 32·32-s − 84·34-s + 40·35-s + 398·37-s + 104·38-s − 40·40-s − 174·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.431·7-s − 0.353·8-s − 0.316·10-s − 0.328·11-s + 0.277·13-s − 0.305·14-s + 1/4·16-s + 0.599·17-s − 0.627·19-s + 0.223·20-s + 0.232·22-s − 1.19·23-s + 1/5·25-s − 0.196·26-s + 0.215·28-s − 1.80·29-s + 0.672·31-s − 0.176·32-s − 0.423·34-s + 0.193·35-s + 1.76·37-s + 0.443·38-s − 0.158·40-s − 0.662·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 13 | \( 1 - p T \) |
| good | 7 | \( 1 - 8 T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 17 | \( 1 - 42 T + p^{3} T^{2} \) |
| 19 | \( 1 + 52 T + p^{3} T^{2} \) |
| 23 | \( 1 + 132 T + p^{3} T^{2} \) |
| 29 | \( 1 + 282 T + p^{3} T^{2} \) |
| 31 | \( 1 - 116 T + p^{3} T^{2} \) |
| 37 | \( 1 - 398 T + p^{3} T^{2} \) |
| 41 | \( 1 + 174 T + p^{3} T^{2} \) |
| 43 | \( 1 + 76 T + p^{3} T^{2} \) |
| 47 | \( 1 + 456 T + p^{3} T^{2} \) |
| 53 | \( 1 + 150 T + p^{3} T^{2} \) |
| 59 | \( 1 - 156 T + p^{3} T^{2} \) |
| 61 | \( 1 - 230 T + p^{3} T^{2} \) |
| 67 | \( 1 + 592 T + p^{3} T^{2} \) |
| 71 | \( 1 + 408 T + p^{3} T^{2} \) |
| 73 | \( 1 + 10 p T + p^{3} T^{2} \) |
| 79 | \( 1 - 728 T + p^{3} T^{2} \) |
| 83 | \( 1 + 36 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1482 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1742 T + p^{3} T^{2} \) |
| 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
−9.041627042746331012254282477322, −8.093915148341754109459487696658, −7.65545557106305667684467755629, −6.44572268853424289517289491891, −5.81264781689457674526103682227, −4.74656100382930563592638150911, −3.54006746123230932425142093362, −2.30422308322889324664419418007, −1.41740232632978912338076923335, 0,
1.41740232632978912338076923335, 2.30422308322889324664419418007, 3.54006746123230932425142093362, 4.74656100382930563592638150911, 5.81264781689457674526103682227, 6.44572268853424289517289491891, 7.65545557106305667684467755629, 8.093915148341754109459487696658, 9.041627042746331012254282477322
