| L(s) = 1 | + 0.741·3-s − 2.48·5-s − 2.45·9-s − 5.32·11-s + 4.45·13-s − 1.84·15-s − 5.22·17-s + 1.97·19-s − 6.97·23-s + 1.18·25-s − 4.04·27-s + 3.94·29-s + 4.03·31-s − 3.95·33-s + 8.44·37-s + 3.30·39-s − 41-s − 10.5·43-s + 6.09·45-s − 4.67·47-s − 3.87·51-s + 2.15·53-s + 13.2·55-s + 1.46·57-s + 3.06·59-s − 12.0·61-s − 11.0·65-s + ⋯ |
| L(s) = 1 | + 0.428·3-s − 1.11·5-s − 0.816·9-s − 1.60·11-s + 1.23·13-s − 0.475·15-s − 1.26·17-s + 0.452·19-s − 1.45·23-s + 0.236·25-s − 0.777·27-s + 0.733·29-s + 0.725·31-s − 0.687·33-s + 1.38·37-s + 0.528·39-s − 0.156·41-s − 1.61·43-s + 0.908·45-s − 0.681·47-s − 0.542·51-s + 0.295·53-s + 1.78·55-s + 0.193·57-s + 0.399·59-s − 1.53·61-s − 1.37·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7963816954\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7963816954\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 0.741T + 3T^{2} \) |
| 5 | \( 1 + 2.48T + 5T^{2} \) |
| 11 | \( 1 + 5.32T + 11T^{2} \) |
| 13 | \( 1 - 4.45T + 13T^{2} \) |
| 17 | \( 1 + 5.22T + 17T^{2} \) |
| 19 | \( 1 - 1.97T + 19T^{2} \) |
| 23 | \( 1 + 6.97T + 23T^{2} \) |
| 29 | \( 1 - 3.94T + 29T^{2} \) |
| 31 | \( 1 - 4.03T + 31T^{2} \) |
| 37 | \( 1 - 8.44T + 37T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 4.67T + 47T^{2} \) |
| 53 | \( 1 - 2.15T + 53T^{2} \) |
| 59 | \( 1 - 3.06T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 + 15.6T + 67T^{2} \) |
| 71 | \( 1 + 0.549T + 71T^{2} \) |
| 73 | \( 1 + 6.10T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 - 3.02T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−7.935449729103507116170436279125, −7.44652671480120668037097432808, −6.32267653930416652853625382180, −5.90461168921145358583769210614, −4.85513644851181608865733623088, −4.27965498217098345191216341384, −3.37912997338195859697619094282, −2.85251477432079314841296651364, −1.92674858046763407640214720333, −0.40745626347520622290016569799,
0.40745626347520622290016569799, 1.92674858046763407640214720333, 2.85251477432079314841296651364, 3.37912997338195859697619094282, 4.27965498217098345191216341384, 4.85513644851181608865733623088, 5.90461168921145358583769210614, 6.32267653930416652853625382180, 7.44652671480120668037097432808, 7.935449729103507116170436279125
