| L(s) = 1 | + 0.0290·3-s + 3.75·5-s − 1.56·7-s − 2.99·9-s − 3.66·11-s − 0.408·13-s + 0.108·15-s + 17-s − 0.566·19-s − 0.0453·21-s + 3.31·23-s + 9.11·25-s − 0.173·27-s + 7.72·29-s − 4.90·31-s − 0.106·33-s − 5.87·35-s + 10.2·37-s − 0.0118·39-s + 5.36·41-s + 1.36·43-s − 11.2·45-s − 0.136·47-s − 4.55·49-s + 0.0290·51-s + 3.01·53-s − 13.7·55-s + ⋯ |
| L(s) = 1 | + 0.0167·3-s + 1.68·5-s − 0.590·7-s − 0.999·9-s − 1.10·11-s − 0.113·13-s + 0.0281·15-s + 0.242·17-s − 0.130·19-s − 0.00989·21-s + 0.691·23-s + 1.82·25-s − 0.0334·27-s + 1.43·29-s − 0.881·31-s − 0.0185·33-s − 0.992·35-s + 1.68·37-s − 0.00189·39-s + 0.838·41-s + 0.208·43-s − 1.67·45-s − 0.0199·47-s − 0.650·49-s + 0.00406·51-s + 0.414·53-s − 1.85·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.209254907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.209254907\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 - 0.0290T + 3T^{2} \) |
| 5 | \( 1 - 3.75T + 5T^{2} \) |
| 7 | \( 1 + 1.56T + 7T^{2} \) |
| 11 | \( 1 + 3.66T + 11T^{2} \) |
| 13 | \( 1 + 0.408T + 13T^{2} \) |
| 19 | \( 1 + 0.566T + 19T^{2} \) |
| 23 | \( 1 - 3.31T + 23T^{2} \) |
| 29 | \( 1 - 7.72T + 29T^{2} \) |
| 31 | \( 1 + 4.90T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 - 5.36T + 41T^{2} \) |
| 43 | \( 1 - 1.36T + 43T^{2} \) |
| 47 | \( 1 + 0.136T + 47T^{2} \) |
| 53 | \( 1 - 3.01T + 53T^{2} \) |
| 61 | \( 1 + 3.46T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 + 1.80T + 71T^{2} \) |
| 73 | \( 1 + 5.19T + 73T^{2} \) |
| 79 | \( 1 + 4.38T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 + 2.17T + 97T^{2} \) |
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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
−7.86208440364716183989096913559, −7.04367273355870206799999046966, −6.17276884603685772209826207640, −5.87571535264423735627874883805, −5.22700715622550013369504056823, −4.49760265741832681209890562940, −3.02826486633194597411883050829, −2.79572330795906573671554227698, −1.94168530948962839386230769921, −0.71341175043433959782099078933,
0.71341175043433959782099078933, 1.94168530948962839386230769921, 2.79572330795906573671554227698, 3.02826486633194597411883050829, 4.49760265741832681209890562940, 5.22700715622550013369504056823, 5.87571535264423735627874883805, 6.17276884603685772209826207640, 7.04367273355870206799999046966, 7.86208440364716183989096913559
