| L(s) = 1 | + 2.78·2-s + 5.74·4-s + 2.97·5-s − 1.34·7-s + 10.4·8-s + 8.26·10-s + 11-s − 3.44·13-s − 3.74·14-s + 17.5·16-s − 4.36·17-s + 19-s + 17.0·20-s + 2.78·22-s − 8.16·23-s + 3.82·25-s − 9.59·26-s − 7.73·28-s + 1.36·29-s + 2.69·31-s + 27.9·32-s − 12.1·34-s − 3.99·35-s − 1.17·37-s + 2.78·38-s + 30.9·40-s + 3.36·41-s + ⋯ |
| L(s) = 1 | + 1.96·2-s + 2.87·4-s + 1.32·5-s − 0.508·7-s + 3.68·8-s + 2.61·10-s + 0.301·11-s − 0.955·13-s − 1.00·14-s + 4.38·16-s − 1.05·17-s + 0.229·19-s + 3.81·20-s + 0.593·22-s − 1.70·23-s + 0.765·25-s − 1.88·26-s − 1.46·28-s + 0.252·29-s + 0.484·31-s + 4.93·32-s − 2.08·34-s − 0.675·35-s − 0.192·37-s + 0.451·38-s + 4.89·40-s + 0.525·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.209029639\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.209029639\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.78T + 2T^{2} \) |
| 5 | \( 1 - 2.97T + 5T^{2} \) |
| 7 | \( 1 + 1.34T + 7T^{2} \) |
| 13 | \( 1 + 3.44T + 13T^{2} \) |
| 17 | \( 1 + 4.36T + 17T^{2} \) |
| 23 | \( 1 + 8.16T + 23T^{2} \) |
| 29 | \( 1 - 1.36T + 29T^{2} \) |
| 31 | \( 1 - 2.69T + 31T^{2} \) |
| 37 | \( 1 + 1.17T + 37T^{2} \) |
| 41 | \( 1 - 3.36T + 41T^{2} \) |
| 43 | \( 1 + 3.18T + 43T^{2} \) |
| 47 | \( 1 + 6.44T + 47T^{2} \) |
| 53 | \( 1 + 9.21T + 53T^{2} \) |
| 59 | \( 1 - 4.84T + 59T^{2} \) |
| 61 | \( 1 - 0.802T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 + 2.41T + 71T^{2} \) |
| 73 | \( 1 + 2.83T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 4.47T + 83T^{2} \) |
| 89 | \( 1 + 8.70T + 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
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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
−9.681251927315359466906468886322, −8.195878666714421716480301860343, −7.13293898248685562074901331292, −6.39824443019152864628303376290, −6.03173679912631214758456579772, −5.10946327475377384449589243192, −4.44853854768485588662607170174, −3.41096969379596577259637059323, −2.42392053261770673650481379131, −1.83299889348916415789579579287,
1.83299889348916415789579579287, 2.42392053261770673650481379131, 3.41096969379596577259637059323, 4.44853854768485588662607170174, 5.10946327475377384449589243192, 6.03173679912631214758456579772, 6.39824443019152864628303376290, 7.13293898248685562074901331292, 8.195878666714421716480301860343, 9.681251927315359466906468886322
