| L(s) = 1 | + 2.47·2-s + 2.11·3-s + 4.12·4-s + 5.23·6-s − 0.973·7-s + 5.25·8-s + 1.48·9-s − 5.38·11-s + 8.72·12-s − 1.99·13-s − 2.40·14-s + 4.74·16-s − 2.04·17-s + 3.66·18-s + 6.20·19-s − 2.05·21-s − 13.3·22-s + 1.93·23-s + 11.1·24-s − 4.94·26-s − 3.21·27-s − 4.01·28-s + 4.81·29-s − 6.64·31-s + 1.24·32-s − 11.3·33-s − 5.05·34-s + ⋯ |
| L(s) = 1 | + 1.74·2-s + 1.22·3-s + 2.06·4-s + 2.13·6-s − 0.367·7-s + 1.85·8-s + 0.493·9-s − 1.62·11-s + 2.51·12-s − 0.553·13-s − 0.643·14-s + 1.18·16-s − 0.495·17-s + 0.863·18-s + 1.42·19-s − 0.449·21-s − 2.83·22-s + 0.404·23-s + 2.26·24-s − 0.968·26-s − 0.618·27-s − 0.758·28-s + 0.894·29-s − 1.19·31-s + 0.220·32-s − 1.98·33-s − 0.866·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.019909342\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.019909342\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 - 2.47T + 2T^{2} \) |
| 3 | \( 1 - 2.11T + 3T^{2} \) |
| 7 | \( 1 + 0.973T + 7T^{2} \) |
| 11 | \( 1 + 5.38T + 11T^{2} \) |
| 13 | \( 1 + 1.99T + 13T^{2} \) |
| 17 | \( 1 + 2.04T + 17T^{2} \) |
| 19 | \( 1 - 6.20T + 19T^{2} \) |
| 23 | \( 1 - 1.93T + 23T^{2} \) |
| 29 | \( 1 - 4.81T + 29T^{2} \) |
| 31 | \( 1 + 6.64T + 31T^{2} \) |
| 37 | \( 1 + 0.978T + 37T^{2} \) |
| 41 | \( 1 - 2.73T + 41T^{2} \) |
| 43 | \( 1 - 3.99T + 43T^{2} \) |
| 47 | \( 1 - 7.21T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 - 6.54T + 59T^{2} \) |
| 61 | \( 1 - 2.72T + 61T^{2} \) |
| 67 | \( 1 - 9.56T + 67T^{2} \) |
| 71 | \( 1 + 5.68T + 71T^{2} \) |
| 73 | \( 1 + 9.35T + 73T^{2} \) |
| 79 | \( 1 + 3.18T + 79T^{2} \) |
| 83 | \( 1 - 6.11T + 83T^{2} \) |
| 89 | \( 1 + 3.00T + 89T^{2} \) |
| 97 | \( 1 - 5.95T + 97T^{2} \) |
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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
−10.78681143398142365053755203135, −9.829647322085932408412563769628, −8.772356975758123212006491959626, −7.63606561959992695305789096284, −7.09121059215793247094879260499, −5.69800670375324317990399832927, −5.03759984494144120275797423567, −3.85034965000340632081014933672, −2.90084945039288553315367856344, −2.38695479368533530070535910153,
2.38695479368533530070535910153, 2.90084945039288553315367856344, 3.85034965000340632081014933672, 5.03759984494144120275797423567, 5.69800670375324317990399832927, 7.09121059215793247094879260499, 7.63606561959992695305789096284, 8.772356975758123212006491959626, 9.829647322085932408412563769628, 10.78681143398142365053755203135
