| L(s) = 1 | + 10.0·3-s + 14.3·5-s + 7·7-s + 74.2·9-s + 11·11-s + 67.8·13-s + 144.·15-s − 17.1·17-s + 13.1·19-s + 70.4·21-s − 109.·23-s + 80.6·25-s + 475.·27-s + 8.40·29-s + 60.6·31-s + 110.·33-s + 100.·35-s − 409.·37-s + 682.·39-s − 438.·41-s − 141.·43-s + 1.06e3·45-s − 446.·47-s + 49·49-s − 172.·51-s − 138.·53-s + 157.·55-s + ⋯ |
| L(s) = 1 | + 1.93·3-s + 1.28·5-s + 0.377·7-s + 2.74·9-s + 0.301·11-s + 1.44·13-s + 2.48·15-s − 0.244·17-s + 0.158·19-s + 0.731·21-s − 0.992·23-s + 0.645·25-s + 3.38·27-s + 0.0537·29-s + 0.351·31-s + 0.583·33-s + 0.484·35-s − 1.81·37-s + 2.80·39-s − 1.67·41-s − 0.500·43-s + 3.52·45-s − 1.38·47-s + 0.142·49-s − 0.474·51-s − 0.359·53-s + 0.386·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.740836656\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.740836656\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 - 11T \) |
| good | 3 | \( 1 - 10.0T + 27T^{2} \) |
| 5 | \( 1 - 14.3T + 125T^{2} \) |
| 13 | \( 1 - 67.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 17.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 13.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 109.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 8.40T + 2.43e4T^{2} \) |
| 31 | \( 1 - 60.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 409.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 438.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 141.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 446.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 138.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 681.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 687.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 584.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 471.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 564.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 175.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 309.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 878.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 103.T + 9.12e5T^{2} \) |
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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
−9.176460989362857156797155722738, −8.584557647575779933935228959337, −8.063077544178472515644198883663, −6.92549374838674315348071241958, −6.18988460456805192189204035704, −4.94264058824071119010017483333, −3.82314871952104351976285773287, −3.09908034623271819435335173175, −1.85791546008814427942007511280, −1.55151143276967361652105575604,
1.55151143276967361652105575604, 1.85791546008814427942007511280, 3.09908034623271819435335173175, 3.82314871952104351976285773287, 4.94264058824071119010017483333, 6.18988460456805192189204035704, 6.92549374838674315348071241958, 8.063077544178472515644198883663, 8.584557647575779933935228959337, 9.176460989362857156797155722738
