| L(s) = 1 | − 9·3-s + 26.3·5-s + 108.·7-s + 81·9-s − 285.·11-s + 169·13-s − 237.·15-s + 964.·17-s − 1.12e3·19-s − 977.·21-s − 3.19e3·23-s − 2.43e3·25-s − 729·27-s − 150.·29-s + 1.01e3·31-s + 2.57e3·33-s + 2.86e3·35-s + 1.13e4·37-s − 1.52e3·39-s + 3.40e3·41-s − 1.28e4·43-s + 2.13e3·45-s − 1.95e4·47-s − 5.01e3·49-s − 8.67e3·51-s − 9.70e3·53-s − 7.52e3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.471·5-s + 0.837·7-s + 0.333·9-s − 0.711·11-s + 0.277·13-s − 0.272·15-s + 0.809·17-s − 0.713·19-s − 0.483·21-s − 1.26·23-s − 0.777·25-s − 0.192·27-s − 0.0332·29-s + 0.190·31-s + 0.410·33-s + 0.394·35-s + 1.35·37-s − 0.160·39-s + 0.315·41-s − 1.05·43-s + 0.157·45-s − 1.29·47-s − 0.298·49-s − 0.467·51-s − 0.474·53-s − 0.335·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 9T \) |
| 13 | \( 1 - 169T \) |
| good | 5 | \( 1 - 26.3T + 3.12e3T^{2} \) |
| 7 | \( 1 - 108.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 285.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 964.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.12e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.19e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 150.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.01e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.13e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.40e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.28e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.95e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.70e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.27e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.14e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.53e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.98e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.76e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.80e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.05e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.65e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.86e4T + 8.58e9T^{2} \) |
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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
−9.685526584328089355798468456868, −8.274042276430446052894306978062, −7.82514455563205165341099015109, −6.52170284928462768188382014341, −5.71776360084898646915058369712, −4.93233942190561517574760165646, −3.87039399529978680144477919958, −2.35252437946620142354283522279, −1.36593773920144369393149596911, 0,
1.36593773920144369393149596911, 2.35252437946620142354283522279, 3.87039399529978680144477919958, 4.93233942190561517574760165646, 5.71776360084898646915058369712, 6.52170284928462768188382014341, 7.82514455563205165341099015109, 8.274042276430446052894306978062, 9.685526584328089355798468456868
