| L(s) = 1 | + 2-s + 0.772·3-s + 4-s + 1.22·5-s + 0.772·6-s + 3.40·7-s + 8-s − 2.40·9-s + 1.22·10-s − 11-s + 0.772·12-s − 1.40·13-s + 3.40·14-s + 0.948·15-s + 16-s − 4.80·17-s − 2.40·18-s − 19-s + 1.22·20-s + 2.62·21-s − 22-s + 6.80·23-s + 0.772·24-s − 3.49·25-s − 1.40·26-s − 4.17·27-s + 3.40·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.446·3-s + 0.5·4-s + 0.548·5-s + 0.315·6-s + 1.28·7-s + 0.353·8-s − 0.800·9-s + 0.388·10-s − 0.301·11-s + 0.223·12-s − 0.389·13-s + 0.909·14-s + 0.244·15-s + 0.250·16-s − 1.16·17-s − 0.566·18-s − 0.229·19-s + 0.274·20-s + 0.573·21-s − 0.213·22-s + 1.41·23-s + 0.157·24-s − 0.698·25-s − 0.275·26-s − 0.803·27-s + 0.643·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.619809670\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.619809670\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 0.772T + 3T^{2} \) |
| 5 | \( 1 - 1.22T + 5T^{2} \) |
| 7 | \( 1 - 3.40T + 7T^{2} \) |
| 13 | \( 1 + 1.40T + 13T^{2} \) |
| 17 | \( 1 + 4.80T + 17T^{2} \) |
| 23 | \( 1 - 6.80T + 23T^{2} \) |
| 29 | \( 1 - 8.03T + 29T^{2} \) |
| 31 | \( 1 + 4.94T + 31T^{2} \) |
| 37 | \( 1 - 2.35T + 37T^{2} \) |
| 41 | \( 1 - 2.94T + 41T^{2} \) |
| 43 | \( 1 + 9.12T + 43T^{2} \) |
| 47 | \( 1 - 5.25T + 47T^{2} \) |
| 53 | \( 1 + 4.45T + 53T^{2} \) |
| 59 | \( 1 + 14.5T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 3.40T + 67T^{2} \) |
| 71 | \( 1 + 7.57T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 + 3.71T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 14.9T + 89T^{2} \) |
| 97 | \( 1 - 2.35T + 97T^{2} \) |
| 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
−11.22934528192453880770479967337, −10.60163410364056655862680837597, −9.250074976273593621794054823911, −8.432071805849707563616367264204, −7.52081400967267636097075807942, −6.33486490588264623934865299008, −5.25052143021557226933800496283, −4.50628536033226800674415192783, −2.93214087583392821062183713820, −1.92206372717885493143257013078,
1.92206372717885493143257013078, 2.93214087583392821062183713820, 4.50628536033226800674415192783, 5.25052143021557226933800496283, 6.33486490588264623934865299008, 7.52081400967267636097075807942, 8.432071805849707563616367264204, 9.250074976273593621794054823911, 10.60163410364056655862680837597, 11.22934528192453880770479967337
