| L(s) = 1 | + 2.28·2-s + 3.23·4-s − 4.23·5-s + 2.82·8-s − 9.69·10-s − 3.70·11-s + 2.28·13-s − 5·17-s − 5.45·19-s − 13.7·20-s − 8.47·22-s + 0.333·23-s + 12.9·25-s + 5.23·26-s − 7.19·29-s + 3.36·31-s − 5.65·32-s − 11.4·34-s − 4.70·37-s − 12.4·38-s − 11.9·40-s + 4.70·41-s + 2.23·43-s − 11.9·44-s + 0.763·46-s + 1.47·47-s + 29.6·50-s + ⋯ |
| L(s) = 1 | + 1.61·2-s + 1.61·4-s − 1.89·5-s + 0.999·8-s − 3.06·10-s − 1.11·11-s + 0.634·13-s − 1.21·17-s − 1.25·19-s − 3.06·20-s − 1.80·22-s + 0.0696·23-s + 2.58·25-s + 1.02·26-s − 1.33·29-s + 0.605·31-s − 1.00·32-s − 1.96·34-s − 0.774·37-s − 2.02·38-s − 1.89·40-s + 0.735·41-s + 0.340·43-s − 1.80·44-s + 0.112·46-s + 0.214·47-s + 4.18·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.28T + 2T^{2} \) |
| 5 | \( 1 + 4.23T + 5T^{2} \) |
| 11 | \( 1 + 3.70T + 11T^{2} \) |
| 13 | \( 1 - 2.28T + 13T^{2} \) |
| 17 | \( 1 + 5T + 17T^{2} \) |
| 19 | \( 1 + 5.45T + 19T^{2} \) |
| 23 | \( 1 - 0.333T + 23T^{2} \) |
| 29 | \( 1 + 7.19T + 29T^{2} \) |
| 31 | \( 1 - 3.36T + 31T^{2} \) |
| 37 | \( 1 + 4.70T + 37T^{2} \) |
| 41 | \( 1 - 4.70T + 41T^{2} \) |
| 43 | \( 1 - 2.23T + 43T^{2} \) |
| 47 | \( 1 - 1.47T + 47T^{2} \) |
| 53 | \( 1 - 3.16T + 53T^{2} \) |
| 59 | \( 1 - 3.94T + 59T^{2} \) |
| 61 | \( 1 + 3.70T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + 0.746T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 - 0.236T + 83T^{2} \) |
| 89 | \( 1 - 1.70T + 89T^{2} \) |
| 97 | \( 1 + 1.95T + 97T^{2} \) |
| 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
−8.931784417484884971309124899599, −8.236903732535624808027300572893, −7.37676777238209853721802696721, −6.68241344916331689584017786868, −5.67620303428959603353526329794, −4.63333547884182629727588088166, −4.16039531196809668671815621033, −3.37552809113310081065867172898, −2.37399114869500803124800628435, 0,
2.37399114869500803124800628435, 3.37552809113310081065867172898, 4.16039531196809668671815621033, 4.63333547884182629727588088166, 5.67620303428959603353526329794, 6.68241344916331689584017786868, 7.37676777238209853721802696721, 8.236903732535624808027300572893, 8.931784417484884971309124899599
