| L(s) = 1 | − 2·2-s − 3.07·3-s + 4·4-s − 1.13·5-s + 6.14·6-s + 31.6·7-s − 8·8-s − 17.5·9-s + 2.27·10-s + 32.0·11-s − 12.2·12-s − 19.4·13-s − 63.2·14-s + 3.49·15-s + 16·16-s − 73.0·17-s + 35.1·18-s − 66.2·19-s − 4.54·20-s − 97.1·21-s − 64.0·22-s + 45.4·23-s + 24.5·24-s − 123.·25-s + 38.8·26-s + 136.·27-s + 126.·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.591·3-s + 0.5·4-s − 0.101·5-s + 0.418·6-s + 1.70·7-s − 0.353·8-s − 0.650·9-s + 0.0718·10-s + 0.877·11-s − 0.295·12-s − 0.414·13-s − 1.20·14-s + 0.0601·15-s + 0.250·16-s − 1.04·17-s + 0.459·18-s − 0.799·19-s − 0.0508·20-s − 1.00·21-s − 0.620·22-s + 0.411·23-s + 0.209·24-s − 0.989·25-s + 0.292·26-s + 0.975·27-s + 0.853·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1922 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1922 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.192666106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.192666106\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 31 | \( 1 \) |
| good | 3 | \( 1 + 3.07T + 27T^{2} \) |
| 5 | \( 1 + 1.13T + 125T^{2} \) |
| 7 | \( 1 - 31.6T + 343T^{2} \) |
| 11 | \( 1 - 32.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 19.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 73.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 66.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 45.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 125.T + 2.43e4T^{2} \) |
| 37 | \( 1 - 316.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 434.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 290.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 267.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 96.6T + 1.48e5T^{2} \) |
| 59 | \( 1 + 434.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 7.78T + 2.26e5T^{2} \) |
| 67 | \( 1 - 31.1T + 3.00e5T^{2} \) |
| 71 | \( 1 + 768.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 153.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 835.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 573.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 774.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.16e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.911038179398625078128376923113, −8.021685889191344093108065282147, −7.52297446928647169982651587823, −6.43340663287014150794321864268, −5.79902797202977251049617036766, −4.77221422894638470058182608327, −4.11178069162428202998976187906, −2.52133824362132049084864453453, −1.69250026023905366086918090881, −0.59018243116033786583108604867,
0.59018243116033786583108604867, 1.69250026023905366086918090881, 2.52133824362132049084864453453, 4.11178069162428202998976187906, 4.77221422894638470058182608327, 5.79902797202977251049617036766, 6.43340663287014150794321864268, 7.52297446928647169982651587823, 8.021685889191344093108065282147, 8.911038179398625078128376923113
