| L(s) = 1 | + 2-s − 0.252·3-s + 4-s − 5-s − 0.252·6-s − 1.23·7-s + 8-s − 2.93·9-s − 10-s + 4.08·11-s − 0.252·12-s + 13-s − 1.23·14-s + 0.252·15-s + 16-s − 1.06·17-s − 2.93·18-s − 8.14·19-s − 20-s + 0.313·21-s + 4.08·22-s + 6.59·23-s − 0.252·24-s + 25-s + 26-s + 1.49·27-s − 1.23·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.145·3-s + 0.5·4-s − 0.447·5-s − 0.103·6-s − 0.468·7-s + 0.353·8-s − 0.978·9-s − 0.316·10-s + 1.23·11-s − 0.0729·12-s + 0.277·13-s − 0.331·14-s + 0.0652·15-s + 0.250·16-s − 0.258·17-s − 0.692·18-s − 1.86·19-s − 0.223·20-s + 0.0683·21-s + 0.871·22-s + 1.37·23-s − 0.0515·24-s + 0.200·25-s + 0.196·26-s + 0.288·27-s − 0.234·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 3 | \( 1 + 0.252T + 3T^{2} \) |
| 7 | \( 1 + 1.23T + 7T^{2} \) |
| 11 | \( 1 - 4.08T + 11T^{2} \) |
| 17 | \( 1 + 1.06T + 17T^{2} \) |
| 19 | \( 1 + 8.14T + 19T^{2} \) |
| 23 | \( 1 - 6.59T + 23T^{2} \) |
| 29 | \( 1 + 1.85T + 29T^{2} \) |
| 37 | \( 1 + 0.537T + 37T^{2} \) |
| 41 | \( 1 + 0.693T + 41T^{2} \) |
| 43 | \( 1 + 12.5T + 43T^{2} \) |
| 47 | \( 1 - 6.79T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 - 4.92T + 59T^{2} \) |
| 61 | \( 1 - 4.53T + 61T^{2} \) |
| 67 | \( 1 + 7.82T + 67T^{2} \) |
| 71 | \( 1 - 2.45T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 - 0.819T + 79T^{2} \) |
| 83 | \( 1 + 16.3T + 83T^{2} \) |
| 89 | \( 1 - 9.66T + 89T^{2} \) |
| 97 | \( 1 + 4.14T + 97T^{2} \) |
| 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
−8.170078421638837116927505585012, −6.98647458196682498481207981099, −6.56319194967936777080819773370, −5.94662626869916676246187891015, −5.00147649416852869800696144857, −4.21507603821217617716466636578, −3.49869889521612133208211770085, −2.73464016407768138782546903729, −1.53844168845950213150010032283, 0,
1.53844168845950213150010032283, 2.73464016407768138782546903729, 3.49869889521612133208211770085, 4.21507603821217617716466636578, 5.00147649416852869800696144857, 5.94662626869916676246187891015, 6.56319194967936777080819773370, 6.98647458196682498481207981099, 8.170078421638837116927505585012
