| L(s) = 1 | − 1.73·5-s + 7-s − 4.73·11-s + 13-s + 2.19·17-s + 7.19·19-s − 3·23-s − 2.00·25-s − 0.464·29-s + 1.19·31-s − 1.73·35-s + 4.19·37-s − 3.46·41-s − 7·43-s − 7.73·47-s + 49-s + 9.92·53-s + 8.19·55-s + 10.3·59-s − 10·61-s − 1.73·65-s − 14.3·67-s + 1.26·71-s − 9.19·73-s − 4.73·77-s − 11.3·79-s + 0.803·83-s + ⋯ |
| L(s) = 1 | − 0.774·5-s + 0.377·7-s − 1.42·11-s + 0.277·13-s + 0.532·17-s + 1.65·19-s − 0.625·23-s − 0.400·25-s − 0.0861·29-s + 0.214·31-s − 0.292·35-s + 0.689·37-s − 0.541·41-s − 1.06·43-s − 1.12·47-s + 0.142·49-s + 1.36·53-s + 1.10·55-s + 1.35·59-s − 1.28·61-s − 0.214·65-s − 1.75·67-s + 0.150·71-s − 1.07·73-s − 0.539·77-s − 1.28·79-s + 0.0882·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 1.73T + 5T^{2} \) |
| 11 | \( 1 + 4.73T + 11T^{2} \) |
| 17 | \( 1 - 2.19T + 17T^{2} \) |
| 19 | \( 1 - 7.19T + 19T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 + 0.464T + 29T^{2} \) |
| 31 | \( 1 - 1.19T + 31T^{2} \) |
| 37 | \( 1 - 4.19T + 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 + 7T + 43T^{2} \) |
| 47 | \( 1 + 7.73T + 47T^{2} \) |
| 53 | \( 1 - 9.92T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 14.3T + 67T^{2} \) |
| 71 | \( 1 - 1.26T + 71T^{2} \) |
| 73 | \( 1 + 9.19T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 0.803T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 - 2.80T + 97T^{2} \) |
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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
−7.999707871184336575170437022828, −7.76255328296783370355831283802, −6.99638639380194799722392052601, −5.79626303552497990683709708201, −5.26665880898826083530694287613, −4.39919426586222085341056702165, −3.45827785826426306683713689530, −2.71043422278831614548573942804, −1.39839987249752754708912214909, 0,
1.39839987249752754708912214909, 2.71043422278831614548573942804, 3.45827785826426306683713689530, 4.39919426586222085341056702165, 5.26665880898826083530694287613, 5.79626303552497990683709708201, 6.99638639380194799722392052601, 7.76255328296783370355831283802, 7.999707871184336575170437022828
