| L(s) = 1 | − 2.70·2-s + 3·3-s − 0.678·4-s − 4.71·5-s − 8.11·6-s − 33.7·7-s + 23.4·8-s + 9·9-s + 12.7·10-s − 11·11-s − 2.03·12-s + 58.7·13-s + 91.2·14-s − 14.1·15-s − 58.1·16-s − 115.·17-s − 24.3·18-s + 19·19-s + 3.19·20-s − 101.·21-s + 29.7·22-s − 188.·23-s + 70.4·24-s − 102.·25-s − 158.·26-s + 27·27-s + 22.8·28-s + ⋯ |
| L(s) = 1 | − 0.956·2-s + 0.577·3-s − 0.0847·4-s − 0.421·5-s − 0.552·6-s − 1.82·7-s + 1.03·8-s + 0.333·9-s + 0.403·10-s − 0.301·11-s − 0.0489·12-s + 1.25·13-s + 1.74·14-s − 0.243·15-s − 0.908·16-s − 1.64·17-s − 0.318·18-s + 0.229·19-s + 0.0357·20-s − 1.05·21-s + 0.288·22-s − 1.70·23-s + 0.599·24-s − 0.821·25-s − 1.19·26-s + 0.192·27-s + 0.154·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5457159542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5457159542\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 + 11T \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 + 2.70T + 8T^{2} \) |
| 5 | \( 1 + 4.71T + 125T^{2} \) |
| 7 | \( 1 + 33.7T + 343T^{2} \) |
| 13 | \( 1 - 58.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 115.T + 4.91e3T^{2} \) |
| 23 | \( 1 + 188.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 180.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 129.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 132.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 283.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 232.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 488.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 555.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 132.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 306.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 89.4T + 3.00e5T^{2} \) |
| 71 | \( 1 - 453.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 613.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 979.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.33e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 803.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.61e3T + 9.12e5T^{2} \) |
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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
−10.13672143801952350953938636450, −9.090648274215548494648120916157, −8.759495386985136272781989942317, −7.77232297747280927456993880909, −6.84563147432029822482199868043, −5.96996627041269426703799078348, −4.22814520671627893702856436251, −3.55726652961332625977487069887, −2.16220435053137599787213651457, −0.47162075064710513122437914262,
0.47162075064710513122437914262, 2.16220435053137599787213651457, 3.55726652961332625977487069887, 4.22814520671627893702856436251, 5.96996627041269426703799078348, 6.84563147432029822482199868043, 7.77232297747280927456993880909, 8.759495386985136272781989942317, 9.090648274215548494648120916157, 10.13672143801952350953938636450
