| L(s) = 1 | − 1.29·2-s − 0.310·4-s − 2.12·5-s − 3.41·7-s + 3.00·8-s + 2.76·10-s − 11-s + 0.0780·13-s + 4.44·14-s − 3.28·16-s − 5.64·17-s + 1.76·19-s + 0.659·20-s + 1.29·22-s + 6.81·23-s − 0.489·25-s − 0.101·26-s + 1.06·28-s − 4.78·29-s + 4.29·31-s − 1.73·32-s + 7.33·34-s + 7.25·35-s − 8.73·37-s − 2.29·38-s − 6.37·40-s + 8.09·41-s + ⋯ |
| L(s) = 1 | − 0.919·2-s − 0.155·4-s − 0.949·5-s − 1.29·7-s + 1.06·8-s + 0.873·10-s − 0.301·11-s + 0.0216·13-s + 1.18·14-s − 0.820·16-s − 1.36·17-s + 0.405·19-s + 0.147·20-s + 0.277·22-s + 1.42·23-s − 0.0978·25-s − 0.0198·26-s + 0.200·28-s − 0.888·29-s + 0.772·31-s − 0.307·32-s + 1.25·34-s + 1.22·35-s − 1.43·37-s − 0.372·38-s − 1.00·40-s + 1.26·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1342314193\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1342314193\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 1.29T + 2T^{2} \) |
| 5 | \( 1 + 2.12T + 5T^{2} \) |
| 7 | \( 1 + 3.41T + 7T^{2} \) |
| 13 | \( 1 - 0.0780T + 13T^{2} \) |
| 17 | \( 1 + 5.64T + 17T^{2} \) |
| 19 | \( 1 - 1.76T + 19T^{2} \) |
| 23 | \( 1 - 6.81T + 23T^{2} \) |
| 29 | \( 1 + 4.78T + 29T^{2} \) |
| 31 | \( 1 - 4.29T + 31T^{2} \) |
| 37 | \( 1 + 8.73T + 37T^{2} \) |
| 41 | \( 1 - 8.09T + 41T^{2} \) |
| 43 | \( 1 - 0.917T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 + 6.99T + 53T^{2} \) |
| 59 | \( 1 - 6.51T + 59T^{2} \) |
| 61 | \( 1 + 9.43T + 61T^{2} \) |
| 67 | \( 1 + 9.49T + 67T^{2} \) |
| 71 | \( 1 - 0.958T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + 4.52T + 79T^{2} \) |
| 83 | \( 1 + 5.81T + 83T^{2} \) |
| 89 | \( 1 + 5.60T + 89T^{2} \) |
| 97 | \( 1 + 8.86T + 97T^{2} \) |
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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
−7.915380994366783437615520446240, −7.18639551257321090201482435768, −6.82090752319302574287630392221, −5.88032221478676506004511633593, −4.87694517338030696669939723414, −4.27238429274058518925746063932, −3.45715935876150248010312040212, −2.73800127199787515167604046817, −1.46268159527635102474576189841, −0.21576205634731815859082877597,
0.21576205634731815859082877597, 1.46268159527635102474576189841, 2.73800127199787515167604046817, 3.45715935876150248010312040212, 4.27238429274058518925746063932, 4.87694517338030696669939723414, 5.88032221478676506004511633593, 6.82090752319302574287630392221, 7.18639551257321090201482435768, 7.915380994366783437615520446240
