| L(s) = 1 | − 2.61·3-s − 1.23·5-s + 2·7-s + 3.85·9-s − 3.23·13-s + 3.23·15-s + 1.61·17-s − 0.854·19-s − 5.23·21-s + 3.23·23-s − 3.47·25-s − 2.23·27-s − 4.47·29-s − 2·31-s − 2.47·35-s + 9.70·37-s + 8.47·39-s + 3.38·41-s − 11.5·43-s − 4.76·45-s − 2.47·47-s − 3·49-s − 4.23·51-s − 10.4·53-s + 2.23·57-s + 6.38·59-s + 6.47·61-s + ⋯ |
| L(s) = 1 | − 1.51·3-s − 0.552·5-s + 0.755·7-s + 1.28·9-s − 0.897·13-s + 0.835·15-s + 0.392·17-s − 0.195·19-s − 1.14·21-s + 0.674·23-s − 0.694·25-s − 0.430·27-s − 0.830·29-s − 0.359·31-s − 0.417·35-s + 1.59·37-s + 1.35·39-s + 0.528·41-s − 1.76·43-s − 0.710·45-s − 0.360·47-s − 0.428·49-s − 0.593·51-s − 1.43·53-s + 0.296·57-s + 0.830·59-s + 0.828·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7553151074\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7553151074\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2.61T + 3T^{2} \) |
| 5 | \( 1 + 1.23T + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 - 1.61T + 17T^{2} \) |
| 19 | \( 1 + 0.854T + 19T^{2} \) |
| 23 | \( 1 - 3.23T + 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 9.70T + 37T^{2} \) |
| 41 | \( 1 - 3.38T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 + 2.47T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 - 6.38T + 59T^{2} \) |
| 61 | \( 1 - 6.47T + 61T^{2} \) |
| 67 | \( 1 - 0.0901T + 67T^{2} \) |
| 71 | \( 1 - 0.763T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 - 6.32T + 83T^{2} \) |
| 89 | \( 1 - 3.09T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−9.394732666464411374227570891017, −8.152100452353656423129835180312, −7.59633390116369086616235710865, −6.75429646742206895419835629351, −5.93493231876623354282234034511, −5.05811441201296786859328302841, −4.65171650062445769599650040151, −3.51226659073175426096000204201, −1.98142789675670840777899883802, −0.62553718956490124072933769419,
0.62553718956490124072933769419, 1.98142789675670840777899883802, 3.51226659073175426096000204201, 4.65171650062445769599650040151, 5.05811441201296786859328302841, 5.93493231876623354282234034511, 6.75429646742206895419835629351, 7.59633390116369086616235710865, 8.152100452353656423129835180312, 9.394732666464411374227570891017
