| L(s) = 1 | + (−1 + 1.41i)3-s − i·5-s + 4.82i·7-s + (−1.00 − 2.82i)9-s − 4.82·11-s − 1.17·13-s + (1.41 + i)15-s − 0.828i·17-s − 6i·19-s + (−6.82 − 4.82i)21-s + 0.828·23-s − 25-s + (5.00 + 1.41i)27-s + 6i·29-s − 2i·31-s + ⋯ |
| L(s) = 1 | + (−0.577 + 0.816i)3-s − 0.447i·5-s + 1.82i·7-s + (−0.333 − 0.942i)9-s − 1.45·11-s − 0.324·13-s + (0.365 + 0.258i)15-s − 0.200i·17-s − 1.37i·19-s + (−1.49 − 1.05i)21-s + 0.172·23-s − 0.200·25-s + (0.962 + 0.272i)27-s + 1.11i·29-s − 0.359i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1 - 1.41i)T \) |
| 5 | \( 1 + iT \) |
| good | 7 | \( 1 - 4.82iT - 7T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 + 1.17T + 13T^{2} \) |
| 17 | \( 1 + 0.828iT - 17T^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 - 0.828T + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 + 2iT - 31T^{2} \) |
| 37 | \( 1 + 6.82T + 37T^{2} \) |
| 41 | \( 1 + 9.65iT - 41T^{2} \) |
| 43 | \( 1 - 1.17iT - 43T^{2} \) |
| 47 | \( 1 + 3.17T + 47T^{2} \) |
| 53 | \( 1 + 7.65iT - 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 + 4.48iT - 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 6iT - 79T^{2} \) |
| 83 | \( 1 - 5.31T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−9.645812733666428517773047953516, −8.957681960230560678512435682091, −8.424153321154012899292450692719, −7.12003611490561403476145127556, −5.96106662217612777163239722611, −5.12251178693461889315439714230, −4.94128209225941397038737784643, −3.23795905039036865832092386929, −2.30989687645947812239959880051, 0,
1.48010923179670432604690697203, 2.88968016490006584835597453052, 4.13958936815996914161471974474, 5.17470266983475655995497612029, 6.18313784298317670521309286599, 7.03834173591666465335768003058, 7.72677272977581703666703798065, 8.138151731897559740460959451231, 9.935287955289420757807758393698
