| L(s) = 1 | + (−1 + i)3-s + (−1 − i)7-s + i·9-s + 4i·11-s + (−4 − 4i)13-s + (4 − 4i)17-s − 4·19-s + 2·21-s + (−5 + 5i)23-s + (−4 − 4i)27-s + 2i·29-s − 8i·31-s + (−4 − 4i)33-s + 8·39-s − 4·41-s + ⋯ |
| L(s) = 1 | + (−0.577 + 0.577i)3-s + (−0.377 − 0.377i)7-s + 0.333i·9-s + 1.20i·11-s + (−1.10 − 1.10i)13-s + (0.970 − 0.970i)17-s − 0.917·19-s + 0.436·21-s + (−1.04 + 1.04i)23-s + (−0.769 − 0.769i)27-s + 0.371i·29-s − 1.43i·31-s + (−0.696 − 0.696i)33-s + 1.28·39-s − 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (1 - i)T - 3iT^{2} \) |
| 7 | \( 1 + (1 + i)T + 7iT^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 + (4 + 4i)T + 13iT^{2} \) |
| 17 | \( 1 + (-4 + 4i)T - 17iT^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + (5 - 5i)T - 23iT^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 + 8iT - 31T^{2} \) |
| 37 | \( 1 - 37iT^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 + (7 - 7i)T - 43iT^{2} \) |
| 47 | \( 1 + (3 + 3i)T + 47iT^{2} \) |
| 53 | \( 1 + (4 + 4i)T + 53iT^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + (-3 - 3i)T + 67iT^{2} \) |
| 71 | \( 1 + 16iT - 71T^{2} \) |
| 73 | \( 1 + (4 + 4i)T + 73iT^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + (-5 + 5i)T - 83iT^{2} \) |
| 89 | \( 1 - 10iT - 89T^{2} \) |
| 97 | \( 1 + (12 - 12i)T - 97iT^{2} \) |
| 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
−9.919772862619133434876702545280, −9.574661109830573488886018331404, −7.88530630239642942368852942648, −7.53018700744890584918938106709, −6.32401712171205911247656397916, −5.22538638667272582894854542515, −4.70776300480667285909616321813, −3.48644310466134657736818162271, −2.13376328614016300035544002000, 0,
1.69971321921887323016799424782, 3.11864059326382587524087302816, 4.28491194584830605190006286174, 5.58471461078796806943041311876, 6.30755155853662791263111575752, 6.89001267097818063520236380118, 8.124811921448576934330753426699, 8.836036840990811156891313534127, 9.823472163734552459492393067794
