L(s) = 1 | + 3·3-s − 5-s + 6·9-s − 11-s − 13-s − 3·15-s + 3·17-s + 8·19-s + 4·23-s + 25-s + 9·27-s − 3·29-s + 6·31-s − 3·33-s + 8·37-s − 3·39-s − 10·41-s + 12·43-s − 6·45-s − 3·47-s + 9·51-s − 12·53-s + 55-s + 24·57-s + 2·61-s + 65-s − 4·67-s + ⋯ |
L(s) = 1 | + 1.73·3-s − 0.447·5-s + 2·9-s − 0.301·11-s − 0.277·13-s − 0.774·15-s + 0.727·17-s + 1.83·19-s + 0.834·23-s + 1/5·25-s + 1.73·27-s − 0.557·29-s + 1.07·31-s − 0.522·33-s + 1.31·37-s − 0.480·39-s − 1.56·41-s + 1.82·43-s − 0.894·45-s − 0.437·47-s + 1.26·51-s − 1.64·53-s + 0.134·55-s + 3.17·57-s + 0.256·61-s + 0.124·65-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.628070488\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.628070488\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−15.76606945067305, −15.44423853846818, −14.76480130531480, −14.34975720847417, −13.93075962484608, −13.23040315347590, −12.92845595150916, −12.10538276078292, −11.63192679117833, −10.84985150084215, −10.06771588531276, −9.583624914251186, −9.211377689830413, −8.444673030973758, −7.909507264581539, −7.499532780430823, −7.060550893444952, −6.049703100713662, −5.154302038175277, −4.530441636354457, −3.740200057843772, −2.989633985347454, −2.855925499393895, −1.702071063334869, −0.8920413177809668,
0.8920413177809668, 1.702071063334869, 2.855925499393895, 2.989633985347454, 3.740200057843772, 4.530441636354457, 5.154302038175277, 6.049703100713662, 7.060550893444952, 7.499532780430823, 7.909507264581539, 8.444673030973758, 9.211377689830413, 9.583624914251186, 10.06771588531276, 10.84985150084215, 11.63192679117833, 12.10538276078292, 12.92845595150916, 13.23040315347590, 13.93075962484608, 14.34975720847417, 14.76480130531480, 15.44423853846818, 15.76606945067305