| L(s) = 1 | + 5.36·2-s + 3·3-s + 20.7·4-s + 8.46·5-s + 16.0·6-s − 8.07·7-s + 68.3·8-s + 9·9-s + 45.3·10-s − 11·11-s + 62.2·12-s + 32.0·13-s − 43.2·14-s + 25.3·15-s + 200.·16-s + 52.4·17-s + 48.2·18-s − 19.0·19-s + 175.·20-s − 24.2·21-s − 58.9·22-s + 16.8·23-s + 205.·24-s − 53.4·25-s + 171.·26-s + 27·27-s − 167.·28-s + ⋯ |
| L(s) = 1 | + 1.89·2-s + 0.577·3-s + 2.59·4-s + 0.756·5-s + 1.09·6-s − 0.436·7-s + 3.02·8-s + 0.333·9-s + 1.43·10-s − 0.301·11-s + 1.49·12-s + 0.682·13-s − 0.826·14-s + 0.436·15-s + 3.13·16-s + 0.747·17-s + 0.631·18-s − 0.229·19-s + 1.96·20-s − 0.251·21-s − 0.571·22-s + 0.152·23-s + 1.74·24-s − 0.427·25-s + 1.29·26-s + 0.192·27-s − 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(12.26672667\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.26672667\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 - 5.36T + 8T^{2} \) |
| 5 | \( 1 - 8.46T + 125T^{2} \) |
| 7 | \( 1 + 8.07T + 343T^{2} \) |
| 13 | \( 1 - 32.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 52.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 19.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 16.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 200.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 179.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 45.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 362.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 48.4T + 7.95e4T^{2} \) |
| 47 | \( 1 + 251.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 209.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 302.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 1.02e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 448.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 216.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 85.7T + 4.93e5T^{2} \) |
| 83 | \( 1 + 120.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 192.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 456.T + 9.12e5T^{2} \) |
| 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
−8.612479121569325460179564446110, −7.82928523505813392666344775160, −6.74391249546016694720176599583, −6.33567422819529255399127052277, −5.47608478897417363467085888167, −4.77056365499975038122434662016, −3.76401507569865812491056825774, −3.09997710238101752256691367478, −2.31655508840415963419778510092, −1.32949326665475195740155755030,
1.32949326665475195740155755030, 2.31655508840415963419778510092, 3.09997710238101752256691367478, 3.76401507569865812491056825774, 4.77056365499975038122434662016, 5.47608478897417363467085888167, 6.33567422819529255399127052277, 6.74391249546016694720176599583, 7.82928523505813392666344775160, 8.612479121569325460179564446110
