| L(s) = 1 | + 9·3-s + 80.6·5-s − 70.5·7-s + 81·9-s − 396.·11-s − 248.·13-s + 725.·15-s + 716.·17-s − 2.64e3·19-s − 634.·21-s + 529·23-s + 3.37e3·25-s + 729·27-s + 6.71e3·29-s + 477.·31-s − 3.57e3·33-s − 5.68e3·35-s + 3.85e3·37-s − 2.23e3·39-s − 1.15e4·41-s − 4.39e3·43-s + 6.52e3·45-s + 4.75e3·47-s − 1.18e4·49-s + 6.44e3·51-s − 9.02e3·53-s − 3.19e4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.44·5-s − 0.544·7-s + 0.333·9-s − 0.988·11-s − 0.408·13-s + 0.832·15-s + 0.601·17-s − 1.68·19-s − 0.314·21-s + 0.208·23-s + 1.07·25-s + 0.192·27-s + 1.48·29-s + 0.0893·31-s − 0.570·33-s − 0.784·35-s + 0.463·37-s − 0.235·39-s − 1.07·41-s − 0.362·43-s + 0.480·45-s + 0.314·47-s − 0.703·49-s + 0.346·51-s − 0.441·53-s − 1.42·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| 23 | \( 1 - 529T \) |
| good | 5 | \( 1 - 80.6T + 3.12e3T^{2} \) |
| 7 | \( 1 + 70.5T + 1.68e4T^{2} \) |
| 11 | \( 1 + 396.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 248.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 716.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.64e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 6.71e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 477.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.85e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.15e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.39e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 4.75e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.02e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.45e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.09e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.62e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 8.06e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.48e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.66e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.30e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.43e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.30e5T + 8.58e9T^{2} \) |
| 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
−8.739039096676937582997839587428, −8.064752656030157047119486777293, −6.90783538273061825731949585304, −6.22521326619874614565223306339, −5.35235606716321377655724337983, −4.42671667064342771322158376101, −3.01431704688580919550737348711, −2.43333876128532342266259466435, −1.46052015619375779446177374069, 0,
1.46052015619375779446177374069, 2.43333876128532342266259466435, 3.01431704688580919550737348711, 4.42671667064342771322158376101, 5.35235606716321377655724337983, 6.22521326619874614565223306339, 6.90783538273061825731949585304, 8.064752656030157047119486777293, 8.739039096676937582997839587428
