| L(s) = 1 | + 3.00·3-s + 5-s − 1.71·7-s + 6.00·9-s + 5.52·11-s − 1.76·13-s + 3.00·15-s + 4.11·17-s − 6.35·19-s − 5.14·21-s + 6.98·23-s + 25-s + 9.02·27-s + 4.92·29-s + 2.00·31-s + 16.5·33-s − 1.71·35-s + 10.6·37-s − 5.29·39-s − 3.91·41-s + 3.13·43-s + 6.00·45-s − 13.2·47-s − 4.05·49-s + 12.3·51-s + 0.449·53-s + 5.52·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s − 0.648·7-s + 2.00·9-s + 1.66·11-s − 0.489·13-s + 0.774·15-s + 0.997·17-s − 1.45·19-s − 1.12·21-s + 1.45·23-s + 0.200·25-s + 1.73·27-s + 0.914·29-s + 0.360·31-s + 2.88·33-s − 0.289·35-s + 1.75·37-s − 0.847·39-s − 0.611·41-s + 0.477·43-s + 0.895·45-s − 1.92·47-s − 0.579·49-s + 1.72·51-s + 0.0617·53-s + 0.744·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.927671654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.927671654\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 - 3.00T + 3T^{2} \) |
| 7 | \( 1 + 1.71T + 7T^{2} \) |
| 11 | \( 1 - 5.52T + 11T^{2} \) |
| 13 | \( 1 + 1.76T + 13T^{2} \) |
| 17 | \( 1 - 4.11T + 17T^{2} \) |
| 19 | \( 1 + 6.35T + 19T^{2} \) |
| 23 | \( 1 - 6.98T + 23T^{2} \) |
| 29 | \( 1 - 4.92T + 29T^{2} \) |
| 31 | \( 1 - 2.00T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 + 3.91T + 41T^{2} \) |
| 43 | \( 1 - 3.13T + 43T^{2} \) |
| 47 | \( 1 + 13.2T + 47T^{2} \) |
| 53 | \( 1 - 0.449T + 53T^{2} \) |
| 59 | \( 1 + 6.20T + 59T^{2} \) |
| 61 | \( 1 - 8.80T + 61T^{2} \) |
| 67 | \( 1 + 8.68T + 67T^{2} \) |
| 71 | \( 1 - 0.504T + 71T^{2} \) |
| 73 | \( 1 + 2.71T + 73T^{2} \) |
| 79 | \( 1 + 3.11T + 79T^{2} \) |
| 83 | \( 1 + 0.154T + 83T^{2} \) |
| 89 | \( 1 + 6.34T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−8.072754145881734122439652171562, −7.06849445155891250022835207646, −6.67848296456095492623084892963, −5.98604171280350329370226975197, −4.69519736341906891149286261553, −4.16929472907195588265874623104, −3.23872513854109935951455348462, −2.87926467439214594323928663285, −1.89179117480792372049938553526, −1.09226093153803127109293926100,
1.09226093153803127109293926100, 1.89179117480792372049938553526, 2.87926467439214594323928663285, 3.23872513854109935951455348462, 4.16929472907195588265874623104, 4.69519736341906891149286261553, 5.98604171280350329370226975197, 6.67848296456095492623084892963, 7.06849445155891250022835207646, 8.072754145881734122439652171562
