| L(s) = 1 | + 5·5-s − 5.26·7-s + 10.2·11-s − 58.5·13-s + 33.8·17-s − 119.·19-s + 182.·23-s + 25·25-s + 49.4·29-s + 200.·31-s − 26.3·35-s − 89.6·37-s + 136.·41-s + 254.·43-s − 61.8·47-s − 315.·49-s − 671.·53-s + 51.3·55-s − 450.·59-s − 126.·61-s − 292.·65-s − 466.·67-s + 370.·71-s + 151.·73-s − 54·77-s − 44.8·79-s − 855.·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.284·7-s + 0.281·11-s − 1.24·13-s + 0.482·17-s − 1.44·19-s + 1.65·23-s + 0.200·25-s + 0.316·29-s + 1.16·31-s − 0.127·35-s − 0.398·37-s + 0.520·41-s + 0.902·43-s − 0.191·47-s − 0.919·49-s − 1.73·53-s + 0.125·55-s − 0.993·59-s − 0.265·61-s − 0.558·65-s − 0.849·67-s + 0.618·71-s + 0.243·73-s − 0.0799·77-s − 0.0638·79-s − 1.13·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 7 | \( 1 + 5.26T + 343T^{2} \) |
| 11 | \( 1 - 10.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 58.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 33.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 119.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 182.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 49.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 200.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 89.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 136.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 254.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 61.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + 671.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 450.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 126.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 466.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 370.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 151.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 44.8T + 4.93e5T^{2} \) |
| 83 | \( 1 + 855.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.09e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.20e3T + 9.12e5T^{2} \) |
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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
−8.379719456637329114812904615988, −7.51263151052315375445297634401, −6.69412684406233751128583113992, −6.10519697897225321301660318974, −5.01422171772442595355341731471, −4.45199419261374205247445681046, −3.16367605458572785439565165001, −2.42945315488577028770780526975, −1.26409144697654621429426973035, 0,
1.26409144697654621429426973035, 2.42945315488577028770780526975, 3.16367605458572785439565165001, 4.45199419261374205247445681046, 5.01422171772442595355341731471, 6.10519697897225321301660318974, 6.69412684406233751128583113992, 7.51263151052315375445297634401, 8.379719456637329114812904615988
