| L(s) = 1 | + 2.42·2-s + 2.86·3-s + 3.89·4-s + 0.116·5-s + 6.94·6-s − 2.69·7-s + 4.58·8-s + 5.19·9-s + 0.281·10-s + 3.38·11-s + 11.1·12-s − 1.05·13-s − 6.55·14-s + 0.332·15-s + 3.35·16-s − 3.65·17-s + 12.5·18-s − 2.27·19-s + 0.451·20-s − 7.72·21-s + 8.21·22-s + 7.49·23-s + 13.1·24-s − 4.98·25-s − 2.55·26-s + 6.26·27-s − 10.5·28-s + ⋯ |
| L(s) = 1 | + 1.71·2-s + 1.65·3-s + 1.94·4-s + 0.0519·5-s + 2.83·6-s − 1.02·7-s + 1.62·8-s + 1.73·9-s + 0.0891·10-s + 1.02·11-s + 3.21·12-s − 0.291·13-s − 1.75·14-s + 0.0857·15-s + 0.838·16-s − 0.886·17-s + 2.96·18-s − 0.523·19-s + 0.100·20-s − 1.68·21-s + 1.75·22-s + 1.56·23-s + 2.68·24-s − 0.997·25-s − 0.500·26-s + 1.20·27-s − 1.98·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1681 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1681 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.476583703\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.476583703\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 41 | \( 1 \) |
| good | 2 | \( 1 - 2.42T + 2T^{2} \) |
| 3 | \( 1 - 2.86T + 3T^{2} \) |
| 5 | \( 1 - 0.116T + 5T^{2} \) |
| 7 | \( 1 + 2.69T + 7T^{2} \) |
| 11 | \( 1 - 3.38T + 11T^{2} \) |
| 13 | \( 1 + 1.05T + 13T^{2} \) |
| 17 | \( 1 + 3.65T + 17T^{2} \) |
| 19 | \( 1 + 2.27T + 19T^{2} \) |
| 23 | \( 1 - 7.49T + 23T^{2} \) |
| 29 | \( 1 + 1.47T + 29T^{2} \) |
| 31 | \( 1 - 3.12T + 31T^{2} \) |
| 37 | \( 1 + 1.12T + 37T^{2} \) |
| 43 | \( 1 - 0.993T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 - 0.542T + 53T^{2} \) |
| 59 | \( 1 - 1.94T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 + 9.43T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 0.635T + 83T^{2} \) |
| 89 | \( 1 - 4.81T + 89T^{2} \) |
| 97 | \( 1 - 2.76T + 97T^{2} \) |
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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
−9.260509874453073049025253507869, −8.658866794495626995690226843504, −7.50337815891227821858259751344, −6.73609196124633168361155773669, −6.25596170791360091268475340325, −4.92645668868729589721160195235, −4.07113931115731241057483899617, −3.45437397103306364049717765146, −2.76295805049801370321544239252, −1.87901163751964285074195741977,
1.87901163751964285074195741977, 2.76295805049801370321544239252, 3.45437397103306364049717765146, 4.07113931115731241057483899617, 4.92645668868729589721160195235, 6.25596170791360091268475340325, 6.73609196124633168361155773669, 7.50337815891227821858259751344, 8.658866794495626995690226843504, 9.260509874453073049025253507869
