| L(s) = 1 | + (1.41 − 1.73i)5-s − 2.82i·7-s − 2i·11-s − 2.82·13-s + 4.89i·17-s − 5.65i·23-s + (−0.999 − 4.89i)25-s − 3.46i·29-s + 3.46·31-s + (−4.89 − 4.00i)35-s − 2.82·37-s − 8·41-s + 9.79·43-s − 1.00·49-s − 8.48·53-s + ⋯ |
| L(s) = 1 | + (0.632 − 0.774i)5-s − 1.06i·7-s − 0.603i·11-s − 0.784·13-s + 1.18i·17-s − 1.17i·23-s + (−0.199 − 0.979i)25-s − 0.643i·29-s + 0.622·31-s + (−0.828 − 0.676i)35-s − 0.464·37-s − 1.24·41-s + 1.49·43-s − 0.142·49-s − 1.16·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.401928474\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.401928474\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.41 + 1.73i)T \) |
| good | 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 - 4.89iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 5.65iT - 23T^{2} \) |
| 29 | \( 1 + 3.46iT - 29T^{2} \) |
| 31 | \( 1 - 3.46T + 31T^{2} \) |
| 37 | \( 1 + 2.82T + 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 - 9.79T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 8.48T + 53T^{2} \) |
| 59 | \( 1 + 10iT - 59T^{2} \) |
| 61 | \( 1 + 6.92iT - 61T^{2} \) |
| 67 | \( 1 + 9.79T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 - 9.79iT - 73T^{2} \) |
| 79 | \( 1 + 3.46T + 79T^{2} \) |
| 83 | \( 1 + 9.79T + 83T^{2} \) |
| 89 | \( 1 + 16T + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−8.351081796627739473466156688171, −7.942650210484084552433285083272, −6.82908484046145897637645933898, −6.23889629739059224170793872713, −5.34079469739926958683309040060, −4.51693693984290961585258234476, −3.85693432006119585591041861078, −2.62213473307540395557881764440, −1.52361737892386495546680109381, −0.42555716391831664027637169803,
1.65337303027636543880810244984, 2.58172830260355756252640801744, 3.14662628856501376741086991740, 4.53172842930631291254476015398, 5.37912394700821099422547748577, 5.88439264973384321812176533068, 7.01979955567468911701412016367, 7.27275280925704754173410998307, 8.376297651606944917883022995426, 9.389972577107795886799693760887
