L(s) = 1 | + 3-s + 3.69·5-s + 0.801·7-s + 9-s + 2.85·11-s + 3.69·15-s + 2.93·17-s + 2.44·19-s + 0.801·21-s + 7.78·23-s + 8.63·25-s + 27-s + 3.85·29-s − 2.34·31-s + 2.85·33-s + 2.96·35-s + 7.44·37-s − 0.850·41-s + 1.61·43-s + 3.69·45-s − 2.44·47-s − 6.35·49-s + 2.93·51-s − 9.96·53-s + 10.5·55-s + 2.44·57-s + 5.38·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.65·5-s + 0.303·7-s + 0.333·9-s + 0.859·11-s + 0.953·15-s + 0.712·17-s + 0.560·19-s + 0.174·21-s + 1.62·23-s + 1.72·25-s + 0.192·27-s + 0.715·29-s − 0.421·31-s + 0.496·33-s + 0.500·35-s + 1.22·37-s − 0.132·41-s + 0.246·43-s + 0.550·45-s − 0.356·47-s − 0.908·49-s + 0.411·51-s − 1.36·53-s + 1.41·55-s + 0.323·57-s + 0.700·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.653813892\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.653813892\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 3.69T + 5T^{2} \) |
| 7 | \( 1 - 0.801T + 7T^{2} \) |
| 11 | \( 1 - 2.85T + 11T^{2} \) |
| 17 | \( 1 - 2.93T + 17T^{2} \) |
| 19 | \( 1 - 2.44T + 19T^{2} \) |
| 23 | \( 1 - 7.78T + 23T^{2} \) |
| 29 | \( 1 - 3.85T + 29T^{2} \) |
| 31 | \( 1 + 2.34T + 31T^{2} \) |
| 37 | \( 1 - 7.44T + 37T^{2} \) |
| 41 | \( 1 + 0.850T + 41T^{2} \) |
| 43 | \( 1 - 1.61T + 43T^{2} \) |
| 47 | \( 1 + 2.44T + 47T^{2} \) |
| 53 | \( 1 + 9.96T + 53T^{2} \) |
| 59 | \( 1 - 5.38T + 59T^{2} \) |
| 61 | \( 1 + 13.2T + 61T^{2} \) |
| 67 | \( 1 + 14.3T + 67T^{2} \) |
| 71 | \( 1 + 8.12T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 + 5.40T + 79T^{2} \) |
| 83 | \( 1 + 7.04T + 83T^{2} \) |
| 89 | \( 1 - 1.13T + 89T^{2} \) |
| 97 | \( 1 + 5.94T + 97T^{2} \) |
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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
−7.77466260867425778780909598637, −7.13787979668869273528889797176, −6.35182893571542793039384006329, −5.84994777525368762836324100569, −5.02392100430033571026840125576, −4.41902091258980306782709716996, −3.17633568766860780801573686458, −2.77503469413040750294299654414, −1.58879115433554025117382291857, −1.23792634125153490387343651101,
1.23792634125153490387343651101, 1.58879115433554025117382291857, 2.77503469413040750294299654414, 3.17633568766860780801573686458, 4.41902091258980306782709716996, 5.02392100430033571026840125576, 5.84994777525368762836324100569, 6.35182893571542793039384006329, 7.13787979668869273528889797176, 7.77466260867425778780909598637