| L(s) = 1 | − 2-s + 3-s + 4-s + 2.59·5-s − 6-s + 2.97·7-s − 8-s + 9-s − 2.59·10-s + 11-s + 12-s − 1.18·13-s − 2.97·14-s + 2.59·15-s + 16-s + 0.0963·17-s − 18-s − 2.56·19-s + 2.59·20-s + 2.97·21-s − 22-s + 3.97·23-s − 24-s + 1.75·25-s + 1.18·26-s + 27-s + 2.97·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.16·5-s − 0.408·6-s + 1.12·7-s − 0.353·8-s + 0.333·9-s − 0.821·10-s + 0.301·11-s + 0.288·12-s − 0.328·13-s − 0.794·14-s + 0.670·15-s + 0.250·16-s + 0.0233·17-s − 0.235·18-s − 0.587·19-s + 0.580·20-s + 0.649·21-s − 0.213·22-s + 0.829·23-s − 0.204·24-s + 0.350·25-s + 0.232·26-s + 0.192·27-s + 0.562·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.599030836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.599030836\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 5 | \( 1 - 2.59T + 5T^{2} \) |
| 7 | \( 1 - 2.97T + 7T^{2} \) |
| 13 | \( 1 + 1.18T + 13T^{2} \) |
| 17 | \( 1 - 0.0963T + 17T^{2} \) |
| 19 | \( 1 + 2.56T + 19T^{2} \) |
| 23 | \( 1 - 3.97T + 23T^{2} \) |
| 29 | \( 1 + 8.81T + 29T^{2} \) |
| 31 | \( 1 - 0.619T + 31T^{2} \) |
| 37 | \( 1 - 6.56T + 37T^{2} \) |
| 41 | \( 1 - 2.93T + 41T^{2} \) |
| 43 | \( 1 - 2.34T + 43T^{2} \) |
| 47 | \( 1 - 5.44T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 4.20T + 59T^{2} \) |
| 67 | \( 1 - 9.47T + 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 - 16.3T + 73T^{2} \) |
| 79 | \( 1 + 0.487T + 79T^{2} \) |
| 83 | \( 1 + 4.89T + 83T^{2} \) |
| 89 | \( 1 + 8.76T + 89T^{2} \) |
| 97 | \( 1 - 4.25T + 97T^{2} \) |
| 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
−8.664958528154956521219340172774, −7.70528702621550125261419464761, −7.27523266957828899751561791830, −6.27133875124510886465315871881, −5.58543947688335960820623108333, −4.74193879246659654641978549071, −3.76288248938424220947095681659, −2.47360455665368736441343339290, −2.00713108192013960301448737916, −1.06658186562614224117357043294,
1.06658186562614224117357043294, 2.00713108192013960301448737916, 2.47360455665368736441343339290, 3.76288248938424220947095681659, 4.74193879246659654641978549071, 5.58543947688335960820623108333, 6.27133875124510886465315871881, 7.27523266957828899751561791830, 7.70528702621550125261419464761, 8.664958528154956521219340172774
