| L(s) = 1 | + 28·3-s + 104·7-s − 1.40e3·9-s + 5.14e3·11-s + 8.60e3·13-s − 2.02e4·17-s − 4.55e4·19-s + 2.91e3·21-s − 7.20e4·23-s − 1.00e5·27-s + 2.31e5·29-s + 8.01e4·31-s + 1.44e5·33-s − 1.04e5·37-s + 2.40e5·39-s + 5.84e5·41-s − 7.95e5·43-s + 4.25e5·47-s − 8.12e5·49-s − 5.67e5·51-s − 1.50e6·53-s − 1.27e6·57-s − 2.46e5·59-s + 8.93e5·61-s − 1.45e5·63-s − 2.33e6·67-s − 2.01e6·69-s + ⋯ |
| L(s) = 1 | + 0.598·3-s + 0.114·7-s − 0.641·9-s + 1.16·11-s + 1.08·13-s − 1.00·17-s − 1.52·19-s + 0.0686·21-s − 1.23·23-s − 0.982·27-s + 1.76·29-s + 0.483·31-s + 0.698·33-s − 0.339·37-s + 0.650·39-s + 1.32·41-s − 1.52·43-s + 0.598·47-s − 0.986·49-s − 0.599·51-s − 1.38·53-s − 0.911·57-s − 0.156·59-s + 0.504·61-s − 0.0735·63-s − 0.949·67-s − 0.739·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 28 T + p^{7} T^{2} \) |
| 7 | \( 1 - 104 T + p^{7} T^{2} \) |
| 11 | \( 1 - 468 p T + p^{7} T^{2} \) |
| 13 | \( 1 - 8602 T + p^{7} T^{2} \) |
| 17 | \( 1 + 20274 T + p^{7} T^{2} \) |
| 19 | \( 1 + 45500 T + p^{7} T^{2} \) |
| 23 | \( 1 + 72072 T + p^{7} T^{2} \) |
| 29 | \( 1 - 231510 T + p^{7} T^{2} \) |
| 31 | \( 1 - 80128 T + p^{7} T^{2} \) |
| 37 | \( 1 + 104654 T + p^{7} T^{2} \) |
| 41 | \( 1 - 584922 T + p^{7} T^{2} \) |
| 43 | \( 1 + 795532 T + p^{7} T^{2} \) |
| 47 | \( 1 - 425664 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1500798 T + p^{7} T^{2} \) |
| 59 | \( 1 + 246420 T + p^{7} T^{2} \) |
| 61 | \( 1 - 893942 T + p^{7} T^{2} \) |
| 67 | \( 1 + 2336836 T + p^{7} T^{2} \) |
| 71 | \( 1 - 203688 T + p^{7} T^{2} \) |
| 73 | \( 1 - 3805702 T + p^{7} T^{2} \) |
| 79 | \( 1 + 5053040 T + p^{7} T^{2} \) |
| 83 | \( 1 + 45492 T + p^{7} T^{2} \) |
| 89 | \( 1 - 980010 T + p^{7} T^{2} \) |
| 97 | \( 1 - 5247646 T + p^{7} T^{2} \) |
| 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
−9.441450388614329715414974908341, −8.555974954358330346690772167388, −8.207554894586783543371184528953, −6.57865830971072091454321198837, −6.14093709637020092893256109245, −4.51974227173329201496532806625, −3.71380321301171909415276486567, −2.50871207218533223629321643990, −1.45254813005975157918131468850, 0,
1.45254813005975157918131468850, 2.50871207218533223629321643990, 3.71380321301171909415276486567, 4.51974227173329201496532806625, 6.14093709637020092893256109245, 6.57865830971072091454321198837, 8.207554894586783543371184528953, 8.555974954358330346690772167388, 9.441450388614329715414974908341
