| L(s) = 1 | − 4.75·2-s + 14.6·4-s − 20.4·5-s + 7·7-s − 31.5·8-s + 97.1·10-s − 11·11-s − 9.41·13-s − 33.2·14-s + 32.8·16-s − 10.5·17-s − 3.48·19-s − 298.·20-s + 52.3·22-s + 50.3·23-s + 292.·25-s + 44.7·26-s + 102.·28-s − 11.5·29-s + 169.·31-s + 95.6·32-s + 50.1·34-s − 142.·35-s − 283.·37-s + 16.5·38-s + 643.·40-s + 165.·41-s + ⋯ |
| L(s) = 1 | − 1.68·2-s + 1.82·4-s − 1.82·5-s + 0.377·7-s − 1.39·8-s + 3.07·10-s − 0.301·11-s − 0.200·13-s − 0.635·14-s + 0.513·16-s − 0.150·17-s − 0.0420·19-s − 3.33·20-s + 0.507·22-s + 0.456·23-s + 2.33·25-s + 0.337·26-s + 0.690·28-s − 0.0740·29-s + 0.980·31-s + 0.528·32-s + 0.252·34-s − 0.690·35-s − 1.26·37-s + 0.0707·38-s + 2.54·40-s + 0.632·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 4.75T + 8T^{2} \) |
| 5 | \( 1 + 20.4T + 125T^{2} \) |
| 13 | \( 1 + 9.41T + 2.19e3T^{2} \) |
| 17 | \( 1 + 10.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 3.48T + 6.85e3T^{2} \) |
| 23 | \( 1 - 50.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 11.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 169.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 283.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 165.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 209.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 604.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 164.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 292.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 202.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 750.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 14.6T + 3.57e5T^{2} \) |
| 73 | \( 1 - 992.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.28e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 722.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 502.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 532.T + 9.12e5T^{2} \) |
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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
−9.463211053748336953497037445927, −8.571295375157695976911780097941, −8.062202053628427376257503165825, −7.38590706211151393438941735572, −6.68904362127853600676274072233, −4.98854542746674434046658137564, −3.86247563273329270094931333360, −2.57021540552829493167698789347, −1.00093846980919717991116973924, 0,
1.00093846980919717991116973924, 2.57021540552829493167698789347, 3.86247563273329270094931333360, 4.98854542746674434046658137564, 6.68904362127853600676274072233, 7.38590706211151393438941735572, 8.062202053628427376257503165825, 8.571295375157695976911780097941, 9.463211053748336953497037445927
