| L(s) = 1 | − 2·2-s − 1.18·3-s + 4·4-s + 16.7·5-s + 2.36·6-s − 8·8-s − 25.6·9-s − 33.4·10-s + 58.9·11-s − 4.72·12-s + 20.9·13-s − 19.7·15-s + 16·16-s − 124.·17-s + 51.2·18-s − 82.7·19-s + 66.9·20-s − 117.·22-s − 23·23-s + 9.44·24-s + 155.·25-s − 41.8·26-s + 62.1·27-s + 25.6·29-s + 39.5·30-s − 20.8·31-s − 32·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.227·3-s + 0.5·4-s + 1.49·5-s + 0.160·6-s − 0.353·8-s − 0.948·9-s − 1.05·10-s + 1.61·11-s − 0.113·12-s + 0.446·13-s − 0.340·15-s + 0.250·16-s − 1.76·17-s + 0.670·18-s − 0.999·19-s + 0.748·20-s − 1.14·22-s − 0.208·23-s + 0.0803·24-s + 1.24·25-s − 0.315·26-s + 0.442·27-s + 0.164·29-s + 0.240·30-s − 0.120·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.829561152\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.829561152\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 + 1.18T + 27T^{2} \) |
| 5 | \( 1 - 16.7T + 125T^{2} \) |
| 11 | \( 1 - 58.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 20.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 124.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 82.7T + 6.85e3T^{2} \) |
| 29 | \( 1 - 25.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 20.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 177.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 103.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 354.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 587.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 683.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 482.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 248.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 727.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.17e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.15e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 645.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 429.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 865.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 344.T + 9.12e5T^{2} \) |
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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
−8.860534540881422302494977330473, −8.255104862958713191219387003310, −6.72891482915625357388365884366, −6.46161300019744974439158896889, −5.88864573168600863379709488904, −4.81174200797675839812189645781, −3.69745934893772555638103633498, −2.37906748086780206390772561847, −1.83701593069100555441304589450, −0.67973201082228093149804364500,
0.67973201082228093149804364500, 1.83701593069100555441304589450, 2.37906748086780206390772561847, 3.69745934893772555638103633498, 4.81174200797675839812189645781, 5.88864573168600863379709488904, 6.46161300019744974439158896889, 6.72891482915625357388365884366, 8.255104862958713191219387003310, 8.860534540881422302494977330473
