| L(s) = 1 | + 2.23·3-s − 3·5-s + 2.00·9-s − 2.23·11-s − 13-s − 6.70·15-s + 2·17-s − 4.47·23-s + 4·25-s − 2.23·27-s + 29-s − 2.23·31-s − 5.00·33-s − 8·37-s − 2.23·39-s − 11.1·43-s − 6.00·45-s + 6.70·47-s − 7·49-s + 4.47·51-s + 53-s + 6.70·55-s − 4.47·59-s − 10·61-s + 3·65-s − 8.94·67-s − 10.0·69-s + ⋯ |
| L(s) = 1 | + 1.29·3-s − 1.34·5-s + 0.666·9-s − 0.674·11-s − 0.277·13-s − 1.73·15-s + 0.485·17-s − 0.932·23-s + 0.800·25-s − 0.430·27-s + 0.185·29-s − 0.401·31-s − 0.870·33-s − 1.31·37-s − 0.358·39-s − 1.70·43-s − 0.894·45-s + 0.978·47-s − 49-s + 0.626·51-s + 0.137·53-s + 0.904·55-s − 0.582·59-s − 1.28·61-s + 0.372·65-s − 1.09·67-s − 1.20·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 2.23T + 3T^{2} \) |
| 5 | \( 1 + 3T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 2.23T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 4.47T + 23T^{2} \) |
| 31 | \( 1 + 2.23T + 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 - 6.70T + 47T^{2} \) |
| 53 | \( 1 - T + 53T^{2} \) |
| 59 | \( 1 + 4.47T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 8.94T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 6.70T + 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
| 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
−8.629386041822481667416006429006, −7.997922563740592455439511425219, −7.67442012537690592413942429864, −6.78025632349400729375456674364, −5.48417006414668250196717802680, −4.47737941359406629356362814890, −3.59359257419267324418086907731, −3.04420383265898697560775184512, −1.89101669863293138152600833055, 0,
1.89101669863293138152600833055, 3.04420383265898697560775184512, 3.59359257419267324418086907731, 4.47737941359406629356362814890, 5.48417006414668250196717802680, 6.78025632349400729375456674364, 7.67442012537690592413942429864, 7.997922563740592455439511425219, 8.629386041822481667416006429006
