| L(s) = 1 | − 2-s − 1.00·3-s + 4-s − 1.74·5-s + 1.00·6-s − 4.04·7-s − 8-s − 1.98·9-s + 1.74·10-s − 4.02·11-s − 1.00·12-s − 3.01·13-s + 4.04·14-s + 1.75·15-s + 16-s − 0.409·17-s + 1.98·18-s + 19-s − 1.74·20-s + 4.06·21-s + 4.02·22-s + 4.34·23-s + 1.00·24-s − 1.96·25-s + 3.01·26-s + 5.02·27-s − 4.04·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.581·3-s + 0.5·4-s − 0.779·5-s + 0.411·6-s − 1.52·7-s − 0.353·8-s − 0.662·9-s + 0.551·10-s − 1.21·11-s − 0.290·12-s − 0.836·13-s + 1.07·14-s + 0.453·15-s + 0.250·16-s − 0.0994·17-s + 0.468·18-s + 0.229·19-s − 0.389·20-s + 0.887·21-s + 0.858·22-s + 0.906·23-s + 0.205·24-s − 0.392·25-s + 0.591·26-s + 0.966·27-s − 0.763·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 + 1.00T + 3T^{2} \) |
| 5 | \( 1 + 1.74T + 5T^{2} \) |
| 7 | \( 1 + 4.04T + 7T^{2} \) |
| 11 | \( 1 + 4.02T + 11T^{2} \) |
| 13 | \( 1 + 3.01T + 13T^{2} \) |
| 17 | \( 1 + 0.409T + 17T^{2} \) |
| 23 | \( 1 - 4.34T + 23T^{2} \) |
| 29 | \( 1 - 1.15T + 29T^{2} \) |
| 31 | \( 1 + 7.13T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 - 7.02T + 41T^{2} \) |
| 43 | \( 1 + 1.02T + 43T^{2} \) |
| 47 | \( 1 - 0.0143T + 47T^{2} \) |
| 53 | \( 1 + 3.62T + 53T^{2} \) |
| 59 | \( 1 - 2.69T + 59T^{2} \) |
| 61 | \( 1 - 3.44T + 61T^{2} \) |
| 67 | \( 1 + 4.17T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 16.7T + 79T^{2} \) |
| 83 | \( 1 + 3.77T + 83T^{2} \) |
| 89 | \( 1 - 5.73T + 89T^{2} \) |
| 97 | \( 1 - 0.191T + 97T^{2} \) |
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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
−7.58790721515565334809573189268, −6.88864644842904597375780398847, −6.17801845262467701063142520533, −5.55709529695695037390888914603, −4.81038923294503449147428943870, −3.72231294890806386212628015640, −2.92948516556257167258217441744, −2.43035323395516521998439862479, −0.66589700744213114955478155673, 0,
0.66589700744213114955478155673, 2.43035323395516521998439862479, 2.92948516556257167258217441744, 3.72231294890806386212628015640, 4.81038923294503449147428943870, 5.55709529695695037390888914603, 6.17801845262467701063142520533, 6.88864644842904597375780398847, 7.58790721515565334809573189268
