| L(s) = 1 | + 2.56·3-s − 5-s − 7-s + 3.56·9-s − 2.56·11-s + 5.68·13-s − 2.56·15-s + 3.43·17-s + 1.12·19-s − 2.56·21-s + 5.12·23-s + 25-s + 1.43·27-s − 4.56·29-s + 10.2·31-s − 6.56·33-s + 35-s − 8.24·37-s + 14.5·39-s + 7.12·41-s + 1.12·43-s − 3.56·45-s − 6.56·47-s + 49-s + 8.80·51-s + 4.87·53-s + 2.56·55-s + ⋯ |
| L(s) = 1 | + 1.47·3-s − 0.447·5-s − 0.377·7-s + 1.18·9-s − 0.772·11-s + 1.57·13-s − 0.661·15-s + 0.833·17-s + 0.257·19-s − 0.558·21-s + 1.06·23-s + 0.200·25-s + 0.276·27-s − 0.847·29-s + 1.84·31-s − 1.14·33-s + 0.169·35-s − 1.35·37-s + 2.33·39-s + 1.11·41-s + 0.171·43-s − 0.530·45-s − 0.957·47-s + 0.142·49-s + 1.23·51-s + 0.669·53-s + 0.345·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.885655904\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.885655904\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - 2.56T + 3T^{2} \) |
| 11 | \( 1 + 2.56T + 11T^{2} \) |
| 13 | \( 1 - 5.68T + 13T^{2} \) |
| 17 | \( 1 - 3.43T + 17T^{2} \) |
| 19 | \( 1 - 1.12T + 19T^{2} \) |
| 23 | \( 1 - 5.12T + 23T^{2} \) |
| 29 | \( 1 + 4.56T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 8.24T + 37T^{2} \) |
| 41 | \( 1 - 7.12T + 41T^{2} \) |
| 43 | \( 1 - 1.12T + 43T^{2} \) |
| 47 | \( 1 + 6.56T + 47T^{2} \) |
| 53 | \( 1 - 4.87T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 15.1T + 61T^{2} \) |
| 67 | \( 1 + 14.2T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + 3.12T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−8.888440225585783614233151823523, −8.273949359172296328764915601089, −7.76186171553721632148810553126, −6.93661388526730255723923729506, −5.95270970983218109135960554127, −4.92335186230312731711724586326, −3.72360740884641937350703114632, −3.32901813046669970016404063607, −2.44132100443288269508840438101, −1.09719484108411733098410469426,
1.09719484108411733098410469426, 2.44132100443288269508840438101, 3.32901813046669970016404063607, 3.72360740884641937350703114632, 4.92335186230312731711724586326, 5.95270970983218109135960554127, 6.93661388526730255723923729506, 7.76186171553721632148810553126, 8.273949359172296328764915601089, 8.888440225585783614233151823523
