| L(s) = 1 | + 3.82·2-s + 6.65·4-s − 5·5-s − 7·7-s − 5.14·8-s − 19.1·10-s − 48.5·11-s − 43.6·13-s − 26.7·14-s − 72.9·16-s + 67.6·17-s − 93.2·19-s − 33.2·20-s − 185.·22-s + 104.·23-s + 25·25-s − 167.·26-s − 46.5·28-s + 58.7·29-s − 9.08·31-s − 238.·32-s + 259.·34-s + 35·35-s − 252.·37-s − 357.·38-s + 25.7·40-s − 276.·41-s + ⋯ |
| L(s) = 1 | + 1.35·2-s + 0.832·4-s − 0.447·5-s − 0.377·7-s − 0.227·8-s − 0.605·10-s − 1.33·11-s − 0.931·13-s − 0.511·14-s − 1.13·16-s + 0.965·17-s − 1.12·19-s − 0.372·20-s − 1.80·22-s + 0.944·23-s + 0.200·25-s − 1.26·26-s − 0.314·28-s + 0.376·29-s − 0.0526·31-s − 1.31·32-s + 1.30·34-s + 0.169·35-s − 1.12·37-s − 1.52·38-s + 0.101·40-s − 1.05·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 - 3.82T + 8T^{2} \) |
| 11 | \( 1 + 48.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 43.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 67.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 93.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 104.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 58.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 9.08T + 2.97e4T^{2} \) |
| 37 | \( 1 + 252.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 276.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 92.6T + 7.95e4T^{2} \) |
| 47 | \( 1 - 582.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 623.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 524.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 352.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 736.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 492.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.16e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 872.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 529.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 385.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 463.T + 9.12e5T^{2} \) |
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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
−10.94134673100186775562682202165, −10.07405441550256819965094553309, −8.793958841921703801403813658371, −7.64082229078550582013608267198, −6.64470175715975278717112869722, −5.42455824634066542670465415236, −4.74262187066406129836835954238, −3.48151172056530155917647529141, −2.53407874209039872863235914006, 0,
2.53407874209039872863235914006, 3.48151172056530155917647529141, 4.74262187066406129836835954238, 5.42455824634066542670465415236, 6.64470175715975278717112869722, 7.64082229078550582013608267198, 8.793958841921703801403813658371, 10.07405441550256819965094553309, 10.94134673100186775562682202165
