| L(s) = 1 | − 2·2-s + 3.64·3-s + 4·4-s − 21.5·5-s − 7.29·6-s + 7·7-s − 8·8-s − 13.6·9-s + 43.0·10-s + 66.9·11-s + 14.5·12-s − 14·14-s − 78.5·15-s + 16·16-s + 21.3·17-s + 27.3·18-s + 160.·19-s − 86.0·20-s + 25.5·21-s − 133.·22-s − 200.·23-s − 29.1·24-s + 337.·25-s − 148.·27-s + 28·28-s − 124.·29-s + 157.·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.702·3-s + 0.5·4-s − 1.92·5-s − 0.496·6-s + 0.377·7-s − 0.353·8-s − 0.506·9-s + 1.36·10-s + 1.83·11-s + 0.351·12-s − 0.267·14-s − 1.35·15-s + 0.250·16-s + 0.304·17-s + 0.358·18-s + 1.93·19-s − 0.962·20-s + 0.265·21-s − 1.29·22-s − 1.81·23-s − 0.248·24-s + 2.70·25-s − 1.05·27-s + 0.188·28-s − 0.798·29-s + 0.955·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.350252082\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.350252082\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 7 | \( 1 - 7T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 3.64T + 27T^{2} \) |
| 5 | \( 1 + 21.5T + 125T^{2} \) |
| 11 | \( 1 - 66.9T + 1.33e3T^{2} \) |
| 17 | \( 1 - 21.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 160.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 200.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 124.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 77.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 33.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 74.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 263.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 297.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 117.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 122.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 137.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 280.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 903.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 353.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 522.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.25e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 299.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.27e3T + 9.12e5T^{2} \) |
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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
−8.509056037652880796739859986048, −7.893793226798394774517245989913, −7.51054808796259188113907984415, −6.64357281822588817439654506913, −5.55794175426686767836781097738, −4.22649970422029898489113168198, −3.68669147916214940864234973259, −2.98436055298408234174689208338, −1.56654689131421309829322693847, −0.58153254357707400889518479843,
0.58153254357707400889518479843, 1.56654689131421309829322693847, 2.98436055298408234174689208338, 3.68669147916214940864234973259, 4.22649970422029898489113168198, 5.55794175426686767836781097738, 6.64357281822588817439654506913, 7.51054808796259188113907984415, 7.893793226798394774517245989913, 8.509056037652880796739859986048
