| L(s) = 1 | + 0.827·3-s − 3.42·5-s + 7-s − 2.31·9-s + 11-s − 13-s − 2.83·15-s − 3.58·17-s + 5.27·19-s + 0.827·21-s + 4.25·23-s + 6.73·25-s − 4.39·27-s − 0.0839·29-s − 9.28·31-s + 0.827·33-s − 3.42·35-s − 7.88·37-s − 0.827·39-s + 6.40·41-s − 3.96·43-s + 7.93·45-s − 4.25·47-s + 49-s − 2.96·51-s + 5.49·53-s − 3.42·55-s + ⋯ |
| L(s) = 1 | + 0.477·3-s − 1.53·5-s + 0.377·7-s − 0.771·9-s + 0.301·11-s − 0.277·13-s − 0.731·15-s − 0.868·17-s + 1.20·19-s + 0.180·21-s + 0.887·23-s + 1.34·25-s − 0.846·27-s − 0.0155·29-s − 1.66·31-s + 0.143·33-s − 0.579·35-s − 1.29·37-s − 0.132·39-s + 1.00·41-s − 0.604·43-s + 1.18·45-s − 0.620·47-s + 0.142·49-s − 0.414·51-s + 0.755·53-s − 0.461·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.201007791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.201007791\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 0.827T + 3T^{2} \) |
| 5 | \( 1 + 3.42T + 5T^{2} \) |
| 17 | \( 1 + 3.58T + 17T^{2} \) |
| 19 | \( 1 - 5.27T + 19T^{2} \) |
| 23 | \( 1 - 4.25T + 23T^{2} \) |
| 29 | \( 1 + 0.0839T + 29T^{2} \) |
| 31 | \( 1 + 9.28T + 31T^{2} \) |
| 37 | \( 1 + 7.88T + 37T^{2} \) |
| 41 | \( 1 - 6.40T + 41T^{2} \) |
| 43 | \( 1 + 3.96T + 43T^{2} \) |
| 47 | \( 1 + 4.25T + 47T^{2} \) |
| 53 | \( 1 - 5.49T + 53T^{2} \) |
| 59 | \( 1 - 7.83T + 59T^{2} \) |
| 61 | \( 1 + 5.37T + 61T^{2} \) |
| 67 | \( 1 + 1.50T + 67T^{2} \) |
| 71 | \( 1 + 3.77T + 71T^{2} \) |
| 73 | \( 1 - 9.42T + 73T^{2} \) |
| 79 | \( 1 + 7.41T + 79T^{2} \) |
| 83 | \( 1 + 7.47T + 83T^{2} \) |
| 89 | \( 1 - 5.44T + 89T^{2} \) |
| 97 | \( 1 + 2.94T + 97T^{2} \) |
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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
−7.74767571431860420840774518253, −7.34891958179856536394231320482, −6.72875124316676459108818936718, −5.57468454723236985078569796414, −4.99407276010444732603318544156, −4.13386154959752057648778736022, −3.49295146187538747979479797373, −2.90510446610828527528006190697, −1.81936192560031063896476822529, −0.51736863516835163225522852267,
0.51736863516835163225522852267, 1.81936192560031063896476822529, 2.90510446610828527528006190697, 3.49295146187538747979479797373, 4.13386154959752057648778736022, 4.99407276010444732603318544156, 5.57468454723236985078569796414, 6.72875124316676459108818936718, 7.34891958179856536394231320482, 7.74767571431860420840774518253
