| L(s) = 1 | + 2-s + 4-s + 5-s + 1.26i·7-s + 8-s + 10-s − 1.70·11-s + 4.06i·13-s + 1.26i·14-s + 16-s − 3.79i·17-s − 5.81·19-s + 20-s − 1.70·22-s − 3.51i·23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447·5-s + 0.478i·7-s + 0.353·8-s + 0.316·10-s − 0.515·11-s + 1.12i·13-s + 0.338i·14-s + 0.250·16-s − 0.920i·17-s − 1.33·19-s + 0.223·20-s − 0.364·22-s − 0.733i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0874 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0874 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.750407402\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.750407402\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (-6.24 - 5.29i)T \) |
| good | 7 | \( 1 - 1.26iT - 7T^{2} \) |
| 11 | \( 1 + 1.70T + 11T^{2} \) |
| 13 | \( 1 - 4.06iT - 13T^{2} \) |
| 17 | \( 1 + 3.79iT - 17T^{2} \) |
| 19 | \( 1 + 5.81T + 19T^{2} \) |
| 23 | \( 1 + 3.51iT - 23T^{2} \) |
| 29 | \( 1 + 1.09iT - 29T^{2} \) |
| 31 | \( 1 - 9.21iT - 31T^{2} \) |
| 37 | \( 1 - 4.05T + 37T^{2} \) |
| 41 | \( 1 - 6.43T + 41T^{2} \) |
| 43 | \( 1 - 3.64iT - 43T^{2} \) |
| 47 | \( 1 - 9.46iT - 47T^{2} \) |
| 53 | \( 1 - 8.27T + 53T^{2} \) |
| 59 | \( 1 - 11.9iT - 59T^{2} \) |
| 61 | \( 1 - 1.02iT - 61T^{2} \) |
| 71 | \( 1 - 6.95iT - 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 - 8.81iT - 79T^{2} \) |
| 83 | \( 1 - 6.03iT - 83T^{2} \) |
| 89 | \( 1 - 2.09iT - 89T^{2} \) |
| 97 | \( 1 - 7.10iT - 97T^{2} \) |
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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
−8.313017119025813035290223717904, −7.27062607395108911437107008588, −6.74834502231456202597637649662, −6.03491800422503775685443583102, −5.40049467232358445594903768961, −4.54615426913170164887900595311, −4.09749061938727830151687333669, −2.66165989431355622085755609057, −2.50209794539154851561634089484, −1.26207497445842473628754592891,
0.51610917552227797468192993910, 1.87356930044935501165143957801, 2.57456440800096501034894516113, 3.60582490528946913351530985160, 4.17507903313300449038076590486, 5.09333274277583766518299453981, 5.81074877051829828917911736209, 6.22931244101681763623624423365, 7.18478427103454876018865264945, 7.83057956299358925010264505484
