| L(s) = 1 | + 0.561·5-s + 2.56·7-s − 5.12·11-s + 13-s − 5.68·17-s − 5.12·19-s − 8·23-s − 4.68·25-s + 2·29-s − 4·31-s + 1.43·35-s + 9.68·37-s + 3.12·41-s − 5.43·43-s − 0.315·47-s − 0.438·49-s − 3.12·53-s − 2.87·55-s + 5.12·59-s + 11.1·61-s + 0.561·65-s + 5.12·67-s − 7.68·71-s − 6·73-s − 13.1·77-s − 8·79-s + 2.24·83-s + ⋯ |
| L(s) = 1 | + 0.251·5-s + 0.968·7-s − 1.54·11-s + 0.277·13-s − 1.37·17-s − 1.17·19-s − 1.66·23-s − 0.936·25-s + 0.371·29-s − 0.718·31-s + 0.243·35-s + 1.59·37-s + 0.487·41-s − 0.829·43-s − 0.0459·47-s − 0.0626·49-s − 0.428·53-s − 0.387·55-s + 0.666·59-s + 1.42·61-s + 0.0696·65-s + 0.625·67-s − 0.912·71-s − 0.702·73-s − 1.49·77-s − 0.900·79-s + 0.246·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 0.561T + 5T^{2} \) |
| 7 | \( 1 - 2.56T + 7T^{2} \) |
| 11 | \( 1 + 5.12T + 11T^{2} \) |
| 17 | \( 1 + 5.68T + 17T^{2} \) |
| 19 | \( 1 + 5.12T + 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 9.68T + 37T^{2} \) |
| 41 | \( 1 - 3.12T + 41T^{2} \) |
| 43 | \( 1 + 5.43T + 43T^{2} \) |
| 47 | \( 1 + 0.315T + 47T^{2} \) |
| 53 | \( 1 + 3.12T + 53T^{2} \) |
| 59 | \( 1 - 5.12T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 - 5.12T + 67T^{2} \) |
| 71 | \( 1 + 7.68T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 2.24T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 8.24T + 97T^{2} \) |
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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
−8.558168308617296678870839936973, −8.189817469536320995498381755327, −7.42939004080712213353533162144, −6.32165813797797317321200663996, −5.64152780024777740854601260775, −4.68767418587151149340530812422, −4.03084740882118211043876039324, −2.49436529164931831797013839525, −1.90088314593070488734462830117, 0,
1.90088314593070488734462830117, 2.49436529164931831797013839525, 4.03084740882118211043876039324, 4.68767418587151149340530812422, 5.64152780024777740854601260775, 6.32165813797797317321200663996, 7.42939004080712213353533162144, 8.189817469536320995498381755327, 8.558168308617296678870839936973
