| L(s) = 1 | − i·3-s + (−2 + i)5-s − 4i·7-s − 9-s − 2·11-s − 4i·13-s + (1 + 2i)15-s − i·17-s − 4·19-s − 4·21-s + 8i·23-s + (3 − 4i)25-s + i·27-s − 4·31-s + 2i·33-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (−0.894 + 0.447i)5-s − 1.51i·7-s − 0.333·9-s − 0.603·11-s − 1.10i·13-s + (0.258 + 0.516i)15-s − 0.242i·17-s − 0.917·19-s − 0.872·21-s + 1.66i·23-s + (0.600 − 0.800i)25-s + 0.192i·27-s − 0.718·31-s + 0.348i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (2 - i)T \) |
| 17 | \( 1 + iT \) |
| good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
| 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
−8.212826491268008396771106511973, −7.67083652431583297224505282198, −7.33049284138953056070409043063, −6.43846978151641949295373502374, −5.43854623969428018910485185104, −4.35437773040819728331564467662, −3.57809701119351855059743537368, −2.72246406162333117044026262022, −1.15577946987589366693044899227, 0,
2.04172834058146469236707575786, 2.92534479791780686538814581570, 4.17358229270245888776486465500, 4.65820738281640509568762940735, 5.63952074148870970505278421595, 6.36265626826201011562031561135, 7.47169772096943907611589042367, 8.372632398913155630490906236790, 8.905614855138918366776491010645
