| L(s) = 1 | − 2.28·2-s + 3.23·4-s − 7-s − 2.82·8-s − 5.99·11-s + 5.47·13-s + 2.28·14-s − 1.95·17-s − 0.763·19-s + 13.7·22-s + 2.49·23-s − 12.5·26-s − 3.23·28-s − 0.540·29-s − 3·31-s + 5.65·32-s + 4.47·34-s − 6.70·37-s + 1.74·38-s + 1.95·41-s + 10.2·43-s − 19.3·44-s − 5.70·46-s + 2.08·47-s + 49-s + 17.7·52-s − 14.0·53-s + ⋯ |
| L(s) = 1 | − 1.61·2-s + 1.61·4-s − 0.377·7-s − 0.999·8-s − 1.80·11-s + 1.51·13-s + 0.611·14-s − 0.474·17-s − 0.175·19-s + 2.92·22-s + 0.520·23-s − 2.45·26-s − 0.611·28-s − 0.100·29-s − 0.538·31-s + 1.00·32-s + 0.766·34-s − 1.10·37-s + 0.283·38-s + 0.305·41-s + 1.56·43-s − 2.92·44-s − 0.841·46-s + 0.303·47-s + 0.142·49-s + 2.45·52-s − 1.93·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5245487907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5245487907\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 + 2.28T + 2T^{2} \) |
| 11 | \( 1 + 5.99T + 11T^{2} \) |
| 13 | \( 1 - 5.47T + 13T^{2} \) |
| 17 | \( 1 + 1.95T + 17T^{2} \) |
| 19 | \( 1 + 0.763T + 19T^{2} \) |
| 23 | \( 1 - 2.49T + 23T^{2} \) |
| 29 | \( 1 + 0.540T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + 6.70T + 37T^{2} \) |
| 41 | \( 1 - 1.95T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 2.08T + 47T^{2} \) |
| 53 | \( 1 + 14.0T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 + 9.70T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 - 1.08T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 2.23T + 79T^{2} \) |
| 83 | \( 1 - 3.16T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 + 5.23T + 97T^{2} \) |
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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
−8.155893776383451653582661576129, −7.955035937630844206239769190992, −7.08210382904580973054747203335, −6.37018363773778563678940560299, −5.61281283990504193364249315068, −4.64289572440631920742306516850, −3.44949875741231211218771234282, −2.58896670155777034773437290572, −1.67421762134005123252944882847, −0.50812155415059351283782559206,
0.50812155415059351283782559206, 1.67421762134005123252944882847, 2.58896670155777034773437290572, 3.44949875741231211218771234282, 4.64289572440631920742306516850, 5.61281283990504193364249315068, 6.37018363773778563678940560299, 7.08210382904580973054747203335, 7.955035937630844206239769190992, 8.155893776383451653582661576129
