| L(s) = 1 | − 0.185·3-s − 0.737·5-s − 7-s − 2.96·9-s − 0.814·11-s − 0.677·13-s + 0.137·15-s + 3.00·17-s + 19-s + 0.185·21-s + 3.24·23-s − 4.45·25-s + 1.10·27-s + 7.57·29-s − 0.853·31-s + 0.151·33-s + 0.737·35-s − 0.870·37-s + 0.125·39-s + 1.42·41-s − 1.10·43-s + 2.18·45-s − 1.87·47-s + 49-s − 0.559·51-s + 5.00·53-s + 0.600·55-s + ⋯ |
| L(s) = 1 | − 0.107·3-s − 0.329·5-s − 0.377·7-s − 0.988·9-s − 0.245·11-s − 0.187·13-s + 0.0353·15-s + 0.729·17-s + 0.229·19-s + 0.0405·21-s + 0.676·23-s − 0.891·25-s + 0.213·27-s + 1.40·29-s − 0.153·31-s + 0.0263·33-s + 0.124·35-s − 0.143·37-s + 0.0201·39-s + 0.223·41-s − 0.168·43-s + 0.325·45-s − 0.273·47-s + 0.142·49-s − 0.0782·51-s + 0.687·53-s + 0.0809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 0.185T + 3T^{2} \) |
| 5 | \( 1 + 0.737T + 5T^{2} \) |
| 11 | \( 1 + 0.814T + 11T^{2} \) |
| 13 | \( 1 + 0.677T + 13T^{2} \) |
| 17 | \( 1 - 3.00T + 17T^{2} \) |
| 23 | \( 1 - 3.24T + 23T^{2} \) |
| 29 | \( 1 - 7.57T + 29T^{2} \) |
| 31 | \( 1 + 0.853T + 31T^{2} \) |
| 37 | \( 1 + 0.870T + 37T^{2} \) |
| 41 | \( 1 - 1.42T + 41T^{2} \) |
| 43 | \( 1 + 1.10T + 43T^{2} \) |
| 47 | \( 1 + 1.87T + 47T^{2} \) |
| 53 | \( 1 - 5.00T + 53T^{2} \) |
| 59 | \( 1 - 6.54T + 59T^{2} \) |
| 61 | \( 1 - 2.00T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 - 8.86T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 + 1.72T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 + 7.37T + 89T^{2} \) |
| 97 | \( 1 + 7.28T + 97T^{2} \) |
| 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
−7.47344265592978470499434958217, −6.76160461643818653939943954086, −5.99951104888279126586183068115, −5.41970013391720671249735557688, −4.71732286207248904467726041822, −3.76197200219744052448254976411, −3.06826506186789931603679496117, −2.40748404145750326418098403700, −1.08506239799303951038936861999, 0,
1.08506239799303951038936861999, 2.40748404145750326418098403700, 3.06826506186789931603679496117, 3.76197200219744052448254976411, 4.71732286207248904467726041822, 5.41970013391720671249735557688, 5.99951104888279126586183068115, 6.76160461643818653939943954086, 7.47344265592978470499434958217
