| L(s) = 1 | − 1.23·5-s + 4.47·7-s − 5.23·11-s + 4.47·13-s + 4·17-s − 5.70·19-s + 23-s − 3.47·25-s + 4.47·29-s + 2.47·31-s − 5.52·35-s + 11.2·37-s + 2·41-s + 4.76·43-s + 4·47-s + 13.0·49-s − 5.23·53-s + 6.47·55-s − 8.94·59-s + 0.763·61-s − 5.52·65-s − 9.70·67-s + 8.94·71-s − 4.47·73-s − 23.4·77-s − 4.47·79-s + 13.2·83-s + ⋯ |
| L(s) = 1 | − 0.552·5-s + 1.69·7-s − 1.57·11-s + 1.24·13-s + 0.970·17-s − 1.30·19-s + 0.208·23-s − 0.694·25-s + 0.830·29-s + 0.444·31-s − 0.934·35-s + 1.84·37-s + 0.312·41-s + 0.726·43-s + 0.583·47-s + 1.85·49-s − 0.719·53-s + 0.872·55-s − 1.16·59-s + 0.0978·61-s − 0.685·65-s − 1.18·67-s + 1.06·71-s − 0.523·73-s − 2.66·77-s − 0.503·79-s + 1.45·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.996696368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.996696368\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 1.23T + 5T^{2} \) |
| 7 | \( 1 - 4.47T + 7T^{2} \) |
| 11 | \( 1 + 5.23T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + 5.70T + 19T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 - 2.47T + 31T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4.76T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 + 5.23T + 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 - 0.763T + 61T^{2} \) |
| 67 | \( 1 + 9.70T + 67T^{2} \) |
| 71 | \( 1 - 8.94T + 71T^{2} \) |
| 73 | \( 1 + 4.47T + 73T^{2} \) |
| 79 | \( 1 + 4.47T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 0.472T + 97T^{2} \) |
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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
−8.255605281152104337350930715300, −8.009680565897890629411444068123, −7.53735962824949295294671174818, −6.22991411436753032640656079097, −5.58274421204864406233438030763, −4.67158363317297495156646533686, −4.17873776784296657915580295329, −2.98604600312621760485585684653, −2.01132445146678338122364783003, −0.871554621531781003664655087636,
0.871554621531781003664655087636, 2.01132445146678338122364783003, 2.98604600312621760485585684653, 4.17873776784296657915580295329, 4.67158363317297495156646533686, 5.58274421204864406233438030763, 6.22991411436753032640656079097, 7.53735962824949295294671174818, 8.009680565897890629411444068123, 8.255605281152104337350930715300
