| L(s) = 1 | + 16.9·3-s + 211.·7-s + 45.4·9-s − 520.·11-s + 732.·13-s − 2.26e3·17-s + 2.03e3·19-s + 3.59e3·21-s + 974.·23-s − 3.35e3·27-s + 5.27e3·29-s − 2.00e3·31-s − 8.84e3·33-s + 3.65e3·37-s + 1.24e4·39-s + 1.71e4·41-s + 4.07e3·43-s + 1.33e4·47-s + 2.80e4·49-s − 3.85e4·51-s + 2.74e4·53-s + 3.45e4·57-s + 3.89e3·59-s + 8.73e3·61-s + 9.61e3·63-s − 4.09e4·67-s + 1.65e4·69-s + ⋯ |
| L(s) = 1 | + 1.08·3-s + 1.63·7-s + 0.186·9-s − 1.29·11-s + 1.20·13-s − 1.90·17-s + 1.29·19-s + 1.77·21-s + 0.384·23-s − 0.885·27-s + 1.16·29-s − 0.374·31-s − 1.41·33-s + 0.438·37-s + 1.30·39-s + 1.58·41-s + 0.335·43-s + 0.880·47-s + 1.66·49-s − 2.07·51-s + 1.34·53-s + 1.40·57-s + 0.145·59-s + 0.300·61-s + 0.305·63-s − 1.11·67-s + 0.418·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.147097358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.147097358\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 16.9T + 243T^{2} \) |
| 7 | \( 1 - 211.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 520.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 732.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.26e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.03e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 974.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.27e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.00e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.65e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.71e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 4.07e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.33e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.74e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.89e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 8.73e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.09e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.68e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 9.82e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.03e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.41e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.93e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.62e4T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.113661392835420541108991255455, −8.655859118421498082161468158621, −7.936007477721221672716299553736, −7.32670352033976206044100147561, −5.89174007550837396153731444395, −4.93814125797901586825394822110, −4.07217802339846389922929701296, −2.79824198781392634931142408473, −2.10418240951584839465770464212, −0.904726333060502513433657607420,
0.904726333060502513433657607420, 2.10418240951584839465770464212, 2.79824198781392634931142408473, 4.07217802339846389922929701296, 4.93814125797901586825394822110, 5.89174007550837396153731444395, 7.32670352033976206044100147561, 7.936007477721221672716299553736, 8.655859118421498082161468158621, 9.113661392835420541108991255455
