| L(s) = 1 | + 2i·7-s − 2·11-s + 2i·17-s − 4·19-s + 2·29-s + 8·31-s − 4i·37-s + 8·41-s − 8i·43-s + 8i·47-s + 3·49-s + 10i·53-s − 6·59-s + 2·61-s + 12i·67-s + ⋯ |
| L(s) = 1 | + 0.755i·7-s − 0.603·11-s + 0.485i·17-s − 0.917·19-s + 0.371·29-s + 1.43·31-s − 0.657i·37-s + 1.24·41-s − 1.21i·43-s + 1.16i·47-s + 0.428·49-s + 1.37i·53-s − 0.781·59-s + 0.256·61-s + 1.46i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7729612428\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7729612428\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−8.263482519997862581872838225219, −7.60156520541526534678197450761, −6.80210727347003648801095748543, −5.95702638777339623935685508467, −5.62423476445626674947342543085, −4.57188879825516906831230876108, −4.05875094117863229498805118068, −2.81002305177037998925185578590, −2.41817320234858887102898167000, −1.22641648879376721981641133002,
0.19504476731858127944305836350, 1.27450346935808381323749123661, 2.43824364946363017935671819789, 3.12777500175668417183418102488, 4.17684028632770771531159683527, 4.64753055910622733873644010225, 5.48792836538713782322122496049, 6.38024546511679715111319236614, 6.86359202801641736529325105487, 7.72774562402193866270981461651
