| L(s) = 1 | + (−1.24 − 1.20i)3-s + 5-s − 4.23i·7-s + (0.103 + 2.99i)9-s − 5.66·11-s − 6.93i·13-s + (−1.24 − 1.20i)15-s + 4.78i·17-s + 1.51·19-s + (−5.09 + 5.26i)21-s − 3.29i·23-s + 25-s + (3.47 − 3.85i)27-s − 10.0i·29-s − 6.69i·31-s + ⋯ |
| L(s) = 1 | + (−0.719 − 0.694i)3-s + 0.447·5-s − 1.59i·7-s + (0.0345 + 0.999i)9-s − 1.70·11-s − 1.92i·13-s + (−0.321 − 0.310i)15-s + 1.15i·17-s + 0.348·19-s + (−1.11 + 1.14i)21-s − 0.686i·23-s + 0.200·25-s + (0.669 − 0.742i)27-s − 1.86i·29-s − 1.20i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.628 - 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.628 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6047913104\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6047913104\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.24 + 1.20i)T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (8.12 + 1.00i)T \) |
| good | 7 | \( 1 + 4.23iT - 7T^{2} \) |
| 11 | \( 1 + 5.66T + 11T^{2} \) |
| 13 | \( 1 + 6.93iT - 13T^{2} \) |
| 17 | \( 1 - 4.78iT - 17T^{2} \) |
| 19 | \( 1 - 1.51T + 19T^{2} \) |
| 23 | \( 1 + 3.29iT - 23T^{2} \) |
| 29 | \( 1 + 10.0iT - 29T^{2} \) |
| 31 | \( 1 + 6.69iT - 31T^{2} \) |
| 37 | \( 1 - 3.64T + 37T^{2} \) |
| 41 | \( 1 + 4.89T + 41T^{2} \) |
| 43 | \( 1 - 5.51iT - 43T^{2} \) |
| 47 | \( 1 - 7.87iT - 47T^{2} \) |
| 53 | \( 1 + 7.35T + 53T^{2} \) |
| 59 | \( 1 - 0.429iT - 59T^{2} \) |
| 61 | \( 1 - 8.83iT - 61T^{2} \) |
| 71 | \( 1 + 10.0iT - 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 + 4.49iT - 79T^{2} \) |
| 83 | \( 1 + 14.7iT - 83T^{2} \) |
| 89 | \( 1 - 4.24iT - 89T^{2} \) |
| 97 | \( 1 - 17.5iT - 97T^{2} \) |
| 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
−7.74340098750782760690241938940, −7.59078036090972108995012121000, −6.25641877102559848647975430042, −5.96566833273369047813237074462, −5.04758587504209640950737207184, −4.35479131982561896077976805141, −3.13981710610329490195252793443, −2.26633945105444183414898865053, −0.972745582197756326014942257853, −0.21877194221562016285395080735,
1.69710011086586075471159469661, 2.64241137919170768685939994690, 3.44044612747346454619596308462, 4.91575915633175013692138283909, 5.07009770700826049799670912444, 5.70962349986219440715231599550, 6.65793900639234659414538447407, 7.21349902098043717316048251375, 8.520648470589602482367092735581, 8.989479153757433435678711141173
