| L(s) = 1 | − 2.73·3-s − 5-s − 3.46·7-s + 4.46·9-s − 0.732·11-s + 13-s + 2.73·15-s + 0.535·17-s + 0.732·19-s + 9.46·21-s + 2.73·23-s + 25-s − 3.99·27-s + 2.53·29-s + 10.1·31-s + 2·33-s + 3.46·35-s − 2.73·39-s + 7.46·41-s − 10.7·43-s − 4.46·45-s + 11.4·47-s + 4.99·49-s − 1.46·51-s − 7.46·53-s + 0.732·55-s − 2·57-s + ⋯ |
| L(s) = 1 | − 1.57·3-s − 0.447·5-s − 1.30·7-s + 1.48·9-s − 0.220·11-s + 0.277·13-s + 0.705·15-s + 0.129·17-s + 0.167·19-s + 2.06·21-s + 0.569·23-s + 0.200·25-s − 0.769·27-s + 0.470·29-s + 1.83·31-s + 0.348·33-s + 0.585·35-s − 0.437·39-s + 1.16·41-s − 1.63·43-s − 0.665·45-s + 1.67·47-s + 0.714·49-s − 0.205·51-s − 1.02·53-s + 0.0987·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5678070994\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5678070994\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 2.73T + 3T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 + 0.732T + 11T^{2} \) |
| 17 | \( 1 - 0.535T + 17T^{2} \) |
| 19 | \( 1 - 0.732T + 19T^{2} \) |
| 23 | \( 1 - 2.73T + 23T^{2} \) |
| 29 | \( 1 - 2.53T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 7.46T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 + 7.46T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 - 8.39T + 61T^{2} \) |
| 67 | \( 1 - 0.928T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 - 0.928T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−10.90408677099640618740331508210, −10.20414253020792576095761356254, −9.368171480062089548096755888861, −8.060210896072692909797751119787, −6.83133973832531209909933199725, −6.34776039853551283174323004610, −5.37276639251637763627303023232, −4.34215244154987436195976911636, −3.04747144414217232239792058212, −0.72301461500310057444827065062,
0.72301461500310057444827065062, 3.04747144414217232239792058212, 4.34215244154987436195976911636, 5.37276639251637763627303023232, 6.34776039853551283174323004610, 6.83133973832531209909933199725, 8.060210896072692909797751119787, 9.368171480062089548096755888861, 10.20414253020792576095761356254, 10.90408677099640618740331508210
