| L(s) = 1 | + 2-s + 3.34·3-s + 4-s − 2.85·5-s + 3.34·6-s − 4.07·7-s + 8-s + 8.21·9-s − 2.85·10-s − 11-s + 3.34·12-s + 5.42·13-s − 4.07·14-s − 9.56·15-s + 16-s − 4.18·17-s + 8.21·18-s − 2.85·20-s − 13.6·21-s − 22-s − 0.463·23-s + 3.34·24-s + 3.16·25-s + 5.42·26-s + 17.4·27-s − 4.07·28-s − 1.75·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.93·3-s + 0.5·4-s − 1.27·5-s + 1.36·6-s − 1.54·7-s + 0.353·8-s + 2.73·9-s − 0.903·10-s − 0.301·11-s + 0.966·12-s + 1.50·13-s − 1.08·14-s − 2.47·15-s + 0.250·16-s − 1.01·17-s + 1.93·18-s − 0.638·20-s − 2.97·21-s − 0.213·22-s − 0.0966·23-s + 0.683·24-s + 0.632·25-s + 1.06·26-s + 3.36·27-s − 0.770·28-s − 0.326·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.939970108\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.939970108\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 3 | \( 1 - 3.34T + 3T^{2} \) |
| 5 | \( 1 + 2.85T + 5T^{2} \) |
| 7 | \( 1 + 4.07T + 7T^{2} \) |
| 13 | \( 1 - 5.42T + 13T^{2} \) |
| 17 | \( 1 + 4.18T + 17T^{2} \) |
| 23 | \( 1 + 0.463T + 23T^{2} \) |
| 29 | \( 1 + 1.75T + 29T^{2} \) |
| 31 | \( 1 - 10.5T + 31T^{2} \) |
| 37 | \( 1 - 3.22T + 37T^{2} \) |
| 41 | \( 1 - 6.15T + 41T^{2} \) |
| 43 | \( 1 - 3.20T + 43T^{2} \) |
| 47 | \( 1 - 5.42T + 47T^{2} \) |
| 53 | \( 1 - 9.50T + 53T^{2} \) |
| 59 | \( 1 - 1.45T + 59T^{2} \) |
| 61 | \( 1 + 3.30T + 61T^{2} \) |
| 67 | \( 1 - 2.34T + 67T^{2} \) |
| 71 | \( 1 + 9.51T + 71T^{2} \) |
| 73 | \( 1 + 8.94T + 73T^{2} \) |
| 79 | \( 1 - 1.26T + 79T^{2} \) |
| 83 | \( 1 - 7.27T + 83T^{2} \) |
| 89 | \( 1 - 1.03T + 89T^{2} \) |
| 97 | \( 1 + 4.04T + 97T^{2} \) |
| 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
−7.80526882607777812229670314854, −7.26852145552725479642490368316, −6.59285098245669256519782173199, −5.96457243611261054270311231586, −4.38445835618258129448836066553, −4.14359686686886990123416119399, −3.46350501887910272306501739125, −2.95882617502264085068441892185, −2.29717216504827459176919734173, −0.900869395537799788108062279763,
0.900869395537799788108062279763, 2.29717216504827459176919734173, 2.95882617502264085068441892185, 3.46350501887910272306501739125, 4.14359686686886990123416119399, 4.38445835618258129448836066553, 5.96457243611261054270311231586, 6.59285098245669256519782173199, 7.26852145552725479642490368316, 7.80526882607777812229670314854
