| L(s) = 1 | + (1.65 − 0.517i)3-s + 5-s + 0.387i·7-s + (2.46 − 1.71i)9-s + 4.30·11-s + 3.73i·13-s + (1.65 − 0.517i)15-s + 3.85i·17-s − 5.95·19-s + (0.200 + 0.640i)21-s + 4.48i·23-s + 25-s + (3.18 − 4.10i)27-s + 7.61i·29-s + 3.91i·31-s + ⋯ |
| L(s) = 1 | + (0.954 − 0.298i)3-s + 0.447·5-s + 0.146i·7-s + (0.821 − 0.570i)9-s + 1.29·11-s + 1.03i·13-s + (0.426 − 0.133i)15-s + 0.934i·17-s − 1.36·19-s + (0.0437 + 0.139i)21-s + 0.934i·23-s + 0.200·25-s + (0.613 − 0.789i)27-s + 1.41i·29-s + 0.702i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.798 - 0.602i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.798 - 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.170407745\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.170407745\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.65 + 0.517i)T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (7.70 - 2.75i)T \) |
| good | 7 | \( 1 - 0.387iT - 7T^{2} \) |
| 11 | \( 1 - 4.30T + 11T^{2} \) |
| 13 | \( 1 - 3.73iT - 13T^{2} \) |
| 17 | \( 1 - 3.85iT - 17T^{2} \) |
| 19 | \( 1 + 5.95T + 19T^{2} \) |
| 23 | \( 1 - 4.48iT - 23T^{2} \) |
| 29 | \( 1 - 7.61iT - 29T^{2} \) |
| 31 | \( 1 - 3.91iT - 31T^{2} \) |
| 37 | \( 1 + 1.55T + 37T^{2} \) |
| 41 | \( 1 - 0.142T + 41T^{2} \) |
| 43 | \( 1 - 8.29iT - 43T^{2} \) |
| 47 | \( 1 - 0.257iT - 47T^{2} \) |
| 53 | \( 1 - 5.19T + 53T^{2} \) |
| 59 | \( 1 + 3.12iT - 59T^{2} \) |
| 61 | \( 1 + 8.47iT - 61T^{2} \) |
| 71 | \( 1 - 0.503iT - 71T^{2} \) |
| 73 | \( 1 - 5.39T + 73T^{2} \) |
| 79 | \( 1 - 2.98iT - 79T^{2} \) |
| 83 | \( 1 + 5.05iT - 83T^{2} \) |
| 89 | \( 1 + 12.6iT - 89T^{2} \) |
| 97 | \( 1 + 3.95iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.779721949742366071027826682849, −7.900095756945161242496656326817, −6.76482777971033244532416598859, −6.70285517105255224328213215492, −5.70579363913297826008293997055, −4.47314590189708648596186843280, −3.90852636758890161365667861317, −3.04818094506090144762860723100, −1.84093491660567347732506599797, −1.47730419500556406789083736147,
0.801702043669997884338592355535, 2.12799271837020426746667569645, 2.72450013422373051789185204092, 3.88108573252880406883995586243, 4.30532059653028657828193492599, 5.34623674995188961148866203119, 6.25324161959430028353202397213, 6.95247853023309586920609504339, 7.73984019034977072753658798475, 8.522594164195851219205032395933
