| L(s) = 1 | + 3-s + 1.69·5-s + 7-s + 9-s − 5.84·11-s + 13-s + 1.69·15-s + 0.964·17-s − 6.58·19-s + 21-s + 7.18·23-s − 2.11·25-s + 27-s + 8.58·29-s + 5.54·31-s − 5.84·33-s + 1.69·35-s + 0.964·37-s + 39-s − 4.88·41-s + 7.18·43-s + 1.69·45-s − 4.73·47-s + 49-s + 0.964·51-s + 13.0·53-s − 9.93·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.759·5-s + 0.377·7-s + 0.333·9-s − 1.76·11-s + 0.277·13-s + 0.438·15-s + 0.233·17-s − 1.51·19-s + 0.218·21-s + 1.49·23-s − 0.422·25-s + 0.192·27-s + 1.59·29-s + 0.996·31-s − 1.01·33-s + 0.287·35-s + 0.158·37-s + 0.160·39-s − 0.762·41-s + 1.09·43-s + 0.253·45-s − 0.690·47-s + 0.142·49-s + 0.134·51-s + 1.78·53-s − 1.33·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.741770188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.741770188\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 1.69T + 5T^{2} \) |
| 11 | \( 1 + 5.84T + 11T^{2} \) |
| 17 | \( 1 - 0.964T + 17T^{2} \) |
| 19 | \( 1 + 6.58T + 19T^{2} \) |
| 23 | \( 1 - 7.18T + 23T^{2} \) |
| 29 | \( 1 - 8.58T + 29T^{2} \) |
| 31 | \( 1 - 5.54T + 31T^{2} \) |
| 37 | \( 1 - 0.964T + 37T^{2} \) |
| 41 | \( 1 + 4.88T + 41T^{2} \) |
| 43 | \( 1 - 7.18T + 43T^{2} \) |
| 47 | \( 1 + 4.73T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 - 4.81T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 4.81T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 7.61T + 79T^{2} \) |
| 83 | \( 1 - 0.133T + 83T^{2} \) |
| 89 | \( 1 - 2.73T + 89T^{2} \) |
| 97 | \( 1 + 5.01T + 97T^{2} \) |
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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
−8.402212451673054161656373253723, −7.83310331233718846993646377124, −6.90966096080823928341009118191, −6.22346275337082751855371536592, −5.25739083034583586050401549370, −4.80427841482444002098961207124, −3.73406822142759606584772722213, −2.56627246969851508750828492399, −2.30490537793436133212303627646, −0.911969848488873060481422557518,
0.911969848488873060481422557518, 2.30490537793436133212303627646, 2.56627246969851508750828492399, 3.73406822142759606584772722213, 4.80427841482444002098961207124, 5.25739083034583586050401549370, 6.22346275337082751855371536592, 6.90966096080823928341009118191, 7.83310331233718846993646377124, 8.402212451673054161656373253723
