| L(s) = 1 | − 4.60·2-s + 13.1·4-s − 1.84·5-s − 7·7-s − 23.9·8-s + 8.49·10-s + 11·11-s + 24.6·13-s + 32.2·14-s + 4.57·16-s − 17.8·17-s + 32.1·19-s − 24.3·20-s − 50.6·22-s − 14.1·23-s − 121.·25-s − 113.·26-s − 92.3·28-s + 41.5·29-s + 175.·31-s + 170.·32-s + 82.3·34-s + 12.9·35-s + 292.·37-s − 147.·38-s + 44.1·40-s − 154.·41-s + ⋯ |
| L(s) = 1 | − 1.62·2-s + 1.64·4-s − 0.164·5-s − 0.377·7-s − 1.05·8-s + 0.268·10-s + 0.301·11-s + 0.525·13-s + 0.615·14-s + 0.0714·16-s − 0.255·17-s + 0.388·19-s − 0.272·20-s − 0.490·22-s − 0.128·23-s − 0.972·25-s − 0.855·26-s − 0.623·28-s + 0.266·29-s + 1.01·31-s + 0.940·32-s + 0.415·34-s + 0.0623·35-s + 1.30·37-s − 0.631·38-s + 0.174·40-s − 0.587·41-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
| \(L(2)\) |
\(\approx\) |
\(0.7034330046\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7034330046\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 4.60T + 8T^{2} \) |
| 5 | \( 1 + 1.84T + 125T^{2} \) |
| 13 | \( 1 - 24.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 17.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 32.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 14.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 41.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 175.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 292.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 154.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 277.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 52.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + 82.3T + 1.48e5T^{2} \) |
| 59 | \( 1 + 712.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 647.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 260.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 369.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.14e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 488.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 548.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 105.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.36e3T + 9.12e5T^{2} \) |
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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
−9.834714463944419357341935098598, −9.300483794889184260812285726285, −8.364852092972904968493954779573, −7.76486902083095688341135759376, −6.76168747437624095766254531347, −6.03114225855345079802739055946, −4.46762749682550905386064060362, −3.12106913975693130489990286922, −1.79913066484315486069267832466, −0.62416324217353957190919860818,
0.62416324217353957190919860818, 1.79913066484315486069267832466, 3.12106913975693130489990286922, 4.46762749682550905386064060362, 6.03114225855345079802739055946, 6.76168747437624095766254531347, 7.76486902083095688341135759376, 8.364852092972904968493954779573, 9.300483794889184260812285726285, 9.834714463944419357341935098598
