| L(s) = 1 | − 5·5-s + 11.8·7-s − 56.2·11-s + 34.5·13-s − 39.2·17-s + 146.·19-s − 23.5·23-s + 25·25-s − 161.·29-s + 29.5·31-s − 59.0·35-s − 217.·37-s − 142.·41-s + 468.·43-s + 394.·47-s − 203.·49-s + 134.·53-s + 281.·55-s − 131.·59-s + 259.·61-s − 172.·65-s − 445.·67-s + 560.·71-s − 88.6·73-s − 663.·77-s − 450.·79-s + 284.·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.637·7-s − 1.54·11-s + 0.738·13-s − 0.560·17-s + 1.76·19-s − 0.213·23-s + 0.200·25-s − 1.03·29-s + 0.171·31-s − 0.285·35-s − 0.967·37-s − 0.541·41-s + 1.65·43-s + 1.22·47-s − 0.593·49-s + 0.349·53-s + 0.689·55-s − 0.289·59-s + 0.545·61-s − 0.330·65-s − 0.811·67-s + 0.937·71-s − 0.142·73-s − 0.982·77-s − 0.641·79-s + 0.375·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 7 | \( 1 - 11.8T + 343T^{2} \) |
| 11 | \( 1 + 56.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 34.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 39.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 146.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 23.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 161.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 29.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 217.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 142.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 468.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 394.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 134.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 131.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 259.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 445.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 560.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 88.6T + 3.89e5T^{2} \) |
| 79 | \( 1 + 450.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 284.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 625.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 193.T + 9.12e5T^{2} \) |
| 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
−8.203911531670464759940125026350, −7.64380663976341507378394020775, −7.00534738883418576579330847413, −5.69961693841825594886236931657, −5.25709546793665623471513887872, −4.27655146000530921664273408744, −3.33407709022172054627372905159, −2.38833519190395985673769927297, −1.20391563311979249016700455810, 0,
1.20391563311979249016700455810, 2.38833519190395985673769927297, 3.33407709022172054627372905159, 4.27655146000530921664273408744, 5.25709546793665623471513887872, 5.69961693841825594886236931657, 7.00534738883418576579330847413, 7.64380663976341507378394020775, 8.203911531670464759940125026350
