| L(s) = 1 | − 2.42·2-s + 3-s + 3.89·4-s − 3.95·5-s − 2.42·6-s − 4.02·7-s − 4.59·8-s + 9-s + 9.59·10-s + 0.425·11-s + 3.89·12-s − 2.68·13-s + 9.76·14-s − 3.95·15-s + 3.36·16-s − 17-s − 2.42·18-s − 1.90·19-s − 15.3·20-s − 4.02·21-s − 1.03·22-s + 4.66·23-s − 4.59·24-s + 10.6·25-s + 6.52·26-s + 27-s − 15.6·28-s + ⋯ |
| L(s) = 1 | − 1.71·2-s + 0.577·3-s + 1.94·4-s − 1.76·5-s − 0.990·6-s − 1.52·7-s − 1.62·8-s + 0.333·9-s + 3.03·10-s + 0.128·11-s + 1.12·12-s − 0.745·13-s + 2.60·14-s − 1.02·15-s + 0.840·16-s − 0.242·17-s − 0.572·18-s − 0.436·19-s − 3.43·20-s − 0.877·21-s − 0.220·22-s + 0.973·23-s − 0.937·24-s + 2.12·25-s + 1.27·26-s + 0.192·27-s − 2.95·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.001487552826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.001487552826\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 + 2.42T + 2T^{2} \) |
| 5 | \( 1 + 3.95T + 5T^{2} \) |
| 7 | \( 1 + 4.02T + 7T^{2} \) |
| 11 | \( 1 - 0.425T + 11T^{2} \) |
| 13 | \( 1 + 2.68T + 13T^{2} \) |
| 19 | \( 1 + 1.90T + 19T^{2} \) |
| 23 | \( 1 - 4.66T + 23T^{2} \) |
| 29 | \( 1 + 2.85T + 29T^{2} \) |
| 31 | \( 1 + 7.16T + 31T^{2} \) |
| 37 | \( 1 + 8.38T + 37T^{2} \) |
| 41 | \( 1 - 2.91T + 41T^{2} \) |
| 43 | \( 1 + 5.62T + 43T^{2} \) |
| 47 | \( 1 + 1.96T + 47T^{2} \) |
| 53 | \( 1 + 4.44T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 + 5.35T + 61T^{2} \) |
| 67 | \( 1 + 4.28T + 67T^{2} \) |
| 71 | \( 1 - 4.91T + 71T^{2} \) |
| 73 | \( 1 + 4.26T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 + 2.68T + 83T^{2} \) |
| 89 | \( 1 + 2.16T + 89T^{2} \) |
| 97 | \( 1 - 6.45T + 97T^{2} \) |
| 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
−7.81523134507616341997795506494, −7.34787135332978363217761961542, −6.96008631602566324119460259697, −6.29182106193811581731722524829, −4.92845105819842562860623706528, −3.94942913377151576257194664879, −3.26829004130877907192974749581, −2.69274317626050245525548860739, −1.47574815668582256742351310672, −0.02412653681007042935887182241,
0.02412653681007042935887182241, 1.47574815668582256742351310672, 2.69274317626050245525548860739, 3.26829004130877907192974749581, 3.94942913377151576257194664879, 4.92845105819842562860623706528, 6.29182106193811581731722524829, 6.96008631602566324119460259697, 7.34787135332978363217761961542, 7.81523134507616341997795506494
