| L(s) = 1 | + 5-s + 1.19·7-s + 3.32·11-s + 1.70·13-s − 6.34·17-s − 1.32·19-s − 6.86·23-s + 25-s + 2.02·29-s + 2.67·31-s + 1.19·35-s + 3.32·37-s + 2.32·41-s + 6.34·43-s + 12.7·47-s − 5.57·49-s + 1.02·53-s + 3.32·55-s + 11.6·59-s − 9.73·61-s + 1.70·65-s + 10.5·67-s + 1.06·71-s + 14.0·73-s + 3.96·77-s − 1.41·79-s + 11.8·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.451·7-s + 1.00·11-s + 0.473·13-s − 1.53·17-s − 0.303·19-s − 1.43·23-s + 0.200·25-s + 0.376·29-s + 0.481·31-s + 0.201·35-s + 0.545·37-s + 0.362·41-s + 0.968·43-s + 1.86·47-s − 0.796·49-s + 0.141·53-s + 0.447·55-s + 1.51·59-s − 1.24·61-s + 0.211·65-s + 1.29·67-s + 0.126·71-s + 1.64·73-s + 0.451·77-s − 0.159·79-s + 1.29·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.427004626\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.427004626\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 - 1.19T + 7T^{2} \) |
| 11 | \( 1 - 3.32T + 11T^{2} \) |
| 13 | \( 1 - 1.70T + 13T^{2} \) |
| 17 | \( 1 + 6.34T + 17T^{2} \) |
| 19 | \( 1 + 1.32T + 19T^{2} \) |
| 23 | \( 1 + 6.86T + 23T^{2} \) |
| 29 | \( 1 - 2.02T + 29T^{2} \) |
| 31 | \( 1 - 2.67T + 31T^{2} \) |
| 37 | \( 1 - 3.32T + 37T^{2} \) |
| 41 | \( 1 - 2.32T + 41T^{2} \) |
| 43 | \( 1 - 6.34T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 - 1.02T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 + 9.73T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 - 1.06T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 + 1.41T + 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 + 11T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−8.127076683307868223153957533930, −7.26688442767580174298093648879, −6.37232033147249678088035253151, −6.17002753300069522167189671801, −5.14276142162837300309740621625, −4.25403695509309653207437152844, −3.87347737813239494746349380792, −2.52357339468073445710010036750, −1.91495136559814539427415706956, −0.821847915870118145921926751260,
0.821847915870118145921926751260, 1.91495136559814539427415706956, 2.52357339468073445710010036750, 3.87347737813239494746349380792, 4.25403695509309653207437152844, 5.14276142162837300309740621625, 6.17002753300069522167189671801, 6.37232033147249678088035253151, 7.26688442767580174298093648879, 8.127076683307868223153957533930
