L(s) = 1 | − 2·4-s − 5-s − 4·13-s + 4·16-s − 3·17-s − 19-s + 2·20-s + 3·23-s + 25-s − 9·29-s + 10·31-s − 4·37-s − 6·41-s + 43-s − 6·47-s + 8·52-s + 9·53-s + 15·59-s − 13·61-s − 8·64-s + 4·65-s + 2·67-s + 6·68-s + 6·71-s − 16·73-s + 2·76-s − 8·79-s + ⋯ |
L(s) = 1 | − 4-s − 0.447·5-s − 1.10·13-s + 16-s − 0.727·17-s − 0.229·19-s + 0.447·20-s + 0.625·23-s + 1/5·25-s − 1.67·29-s + 1.79·31-s − 0.657·37-s − 0.937·41-s + 0.152·43-s − 0.875·47-s + 1.10·52-s + 1.23·53-s + 1.95·59-s − 1.66·61-s − 64-s + 0.496·65-s + 0.244·67-s + 0.727·68-s + 0.712·71-s − 1.87·73-s + 0.229·76-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 266805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 266805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−13.19541150435068, −12.89174081530187, −12.32967269929211, −11.94245415239728, −11.47342959341575, −10.98215895269800, −10.38076113247746, −9.907648657591535, −9.673873042692958, −8.960242438333637, −8.638201441544988, −8.281562712544547, −7.587294679583013, −7.275768950885852, −6.705898927878733, −6.185039332204022, −5.342726830736484, −5.189233978000422, −4.589924102153227, −4.086841102025077, −3.732493838872070, −2.917706165192269, −2.551139920728809, −1.683210331814316, −1.043152472322681, 0, 0,
1.043152472322681, 1.683210331814316, 2.551139920728809, 2.917706165192269, 3.732493838872070, 4.086841102025077, 4.589924102153227, 5.189233978000422, 5.342726830736484, 6.185039332204022, 6.705898927878733, 7.275768950885852, 7.587294679583013, 8.281562712544547, 8.638201441544988, 8.960242438333637, 9.673873042692958, 9.907648657591535, 10.38076113247746, 10.98215895269800, 11.47342959341575, 11.94245415239728, 12.32967269929211, 12.89174081530187, 13.19541150435068