| L(s) = 1 | + 2·5-s + 4·7-s − 4·11-s + 2·13-s + 6·17-s − 4·19-s − 25-s + 2·29-s − 4·31-s + 8·35-s + 2·37-s − 2·41-s + 4·43-s + 8·47-s + 9·49-s + 10·53-s − 8·55-s + 4·59-s − 6·61-s + 4·65-s + 4·67-s − 16·71-s − 6·73-s − 16·77-s − 4·79-s − 12·83-s + 12·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.51·7-s − 1.20·11-s + 0.554·13-s + 1.45·17-s − 0.917·19-s − 1/5·25-s + 0.371·29-s − 0.718·31-s + 1.35·35-s + 0.328·37-s − 0.312·41-s + 0.609·43-s + 1.16·47-s + 9/7·49-s + 1.37·53-s − 1.07·55-s + 0.520·59-s − 0.768·61-s + 0.496·65-s + 0.488·67-s − 1.89·71-s − 0.702·73-s − 1.82·77-s − 0.450·79-s − 1.31·83-s + 1.30·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.912920987\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.912920987\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−10.59963279276057209057832448649, −10.08615312505774861350749378685, −8.844581178668094717437191348011, −8.093022628739907221893201157530, −7.33135946825865482348621844071, −5.82131382158921014822362291005, −5.37335304530331906156568812203, −4.20875443528614711634884665822, −2.59447091927996366543258857368, −1.47354540780337201919328149138,
1.47354540780337201919328149138, 2.59447091927996366543258857368, 4.20875443528614711634884665822, 5.37335304530331906156568812203, 5.82131382158921014822362291005, 7.33135946825865482348621844071, 8.093022628739907221893201157530, 8.844581178668094717437191348011, 10.08615312505774861350749378685, 10.59963279276057209057832448649
