| L(s) = 1 | − 2.75·2-s − 0.402·4-s − 0.313·5-s + 28.5·7-s + 23.1·8-s + 0.863·10-s − 63.2·11-s − 78.7·14-s − 60.6·16-s − 98.8·17-s − 14.4·19-s + 0.126·20-s + 174.·22-s + 14.3·23-s − 124.·25-s − 11.5·28-s + 196.·29-s + 118.·31-s − 18.2·32-s + 272.·34-s − 8.94·35-s + 319.·37-s + 39.9·38-s − 7.25·40-s − 346.·41-s − 69.4·43-s + 25.4·44-s + ⋯ |
| L(s) = 1 | − 0.974·2-s − 0.0503·4-s − 0.0280·5-s + 1.54·7-s + 1.02·8-s + 0.0272·10-s − 1.73·11-s − 1.50·14-s − 0.947·16-s − 1.40·17-s − 0.174·19-s + 0.00141·20-s + 1.68·22-s + 0.130·23-s − 0.999·25-s − 0.0776·28-s + 1.25·29-s + 0.687·31-s − 0.100·32-s + 1.37·34-s − 0.0432·35-s + 1.41·37-s + 0.170·38-s − 0.0286·40-s − 1.31·41-s − 0.246·43-s + 0.0872·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9027427686\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9027427686\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.75T + 8T^{2} \) |
| 5 | \( 1 + 0.313T + 125T^{2} \) |
| 7 | \( 1 - 28.5T + 343T^{2} \) |
| 11 | \( 1 + 63.2T + 1.33e3T^{2} \) |
| 17 | \( 1 + 98.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 14.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 14.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 196.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 118.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 319.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 346.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 69.4T + 7.95e4T^{2} \) |
| 47 | \( 1 - 101.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 594.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 204.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 215.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 68.6T + 3.00e5T^{2} \) |
| 71 | \( 1 + 946.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 779.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 240.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 855.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.26e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 662.T + 9.12e5T^{2} \) |
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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
−8.792040024361805965521131670662, −8.380625010876465931197625968469, −7.80172569296715166805641611266, −7.08188912355695665698204681016, −5.74035659858340926880361001631, −4.75126237305562480039273117922, −4.40225964069154970726104231967, −2.59924426086718583736948297908, −1.76765274860768698619404287873, −0.54058540867024542797197543457,
0.54058540867024542797197543457, 1.76765274860768698619404287873, 2.59924426086718583736948297908, 4.40225964069154970726104231967, 4.75126237305562480039273117922, 5.74035659858340926880361001631, 7.08188912355695665698204681016, 7.80172569296715166805641611266, 8.380625010876465931197625968469, 8.792040024361805965521131670662
