| L(s) = 1 | − 0.302·3-s + 3.30·5-s − 2.90·9-s + 4.60·11-s + 6.60·13-s − 1.00·15-s + 17-s + 6·19-s − 2.60·23-s + 5.90·25-s + 1.78·27-s − 8.60·29-s + 6.69·31-s − 1.39·33-s + 7.21·37-s − 2.00·39-s − 9.51·41-s − 4.30·43-s − 9.60·45-s − 10·47-s − 0.302·51-s + 0.697·53-s + 15.2·55-s − 1.81·57-s − 5.21·59-s + 4.30·61-s + 21.8·65-s + ⋯ |
| L(s) = 1 | − 0.174·3-s + 1.47·5-s − 0.969·9-s + 1.38·11-s + 1.83·13-s − 0.258·15-s + 0.242·17-s + 1.37·19-s − 0.543·23-s + 1.18·25-s + 0.344·27-s − 1.59·29-s + 1.20·31-s − 0.242·33-s + 1.18·37-s − 0.320·39-s − 1.48·41-s − 0.656·43-s − 1.43·45-s − 1.45·47-s − 0.0423·51-s + 0.0957·53-s + 2.05·55-s − 0.240·57-s − 0.678·59-s + 0.550·61-s + 2.70·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.745331713\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.745331713\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 0.302T + 3T^{2} \) |
| 5 | \( 1 - 3.30T + 5T^{2} \) |
| 11 | \( 1 - 4.60T + 11T^{2} \) |
| 13 | \( 1 - 6.60T + 13T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 2.60T + 23T^{2} \) |
| 29 | \( 1 + 8.60T + 29T^{2} \) |
| 31 | \( 1 - 6.69T + 31T^{2} \) |
| 37 | \( 1 - 7.21T + 37T^{2} \) |
| 41 | \( 1 + 9.51T + 41T^{2} \) |
| 43 | \( 1 + 4.30T + 43T^{2} \) |
| 47 | \( 1 + 10T + 47T^{2} \) |
| 53 | \( 1 - 0.697T + 53T^{2} \) |
| 59 | \( 1 + 5.21T + 59T^{2} \) |
| 61 | \( 1 - 4.30T + 61T^{2} \) |
| 67 | \( 1 - 2.69T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - 13.5T + 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 + 9.21T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 2.09T + 97T^{2} \) |
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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
−8.682387309683576626356959481357, −8.108448796978046193475231524049, −6.83359675553433149076650620813, −6.16257719436519209809635571122, −5.84620362911401061994813275465, −5.04154532832702039834195383205, −3.77203565043264282322181540255, −3.10613542763688161550250061349, −1.82186369562600974388642463745, −1.11457962680886830333445368701,
1.11457962680886830333445368701, 1.82186369562600974388642463745, 3.10613542763688161550250061349, 3.77203565043264282322181540255, 5.04154532832702039834195383205, 5.84620362911401061994813275465, 6.16257719436519209809635571122, 6.83359675553433149076650620813, 8.108448796978046193475231524049, 8.682387309683576626356959481357
