| L(s) = 1 | + 0.784·3-s − 2.25·5-s − 7-s − 2.38·9-s + 4.87·11-s − 0.955·13-s − 1.76·15-s + 3.33·19-s − 0.784·21-s − 7.47·23-s + 0.0627·25-s − 4.22·27-s + 0.345·29-s + 2.60·31-s + 3.82·33-s + 2.25·35-s + 2.02·37-s − 0.749·39-s − 4.82·41-s − 2.71·43-s + 5.36·45-s + 6.20·47-s + 49-s + 1.39·53-s − 10.9·55-s + 2.61·57-s − 9.23·59-s + ⋯ |
| L(s) = 1 | + 0.452·3-s − 1.00·5-s − 0.377·7-s − 0.794·9-s + 1.46·11-s − 0.265·13-s − 0.455·15-s + 0.764·19-s − 0.171·21-s − 1.55·23-s + 0.0125·25-s − 0.812·27-s + 0.0641·29-s + 0.467·31-s + 0.665·33-s + 0.380·35-s + 0.332·37-s − 0.120·39-s − 0.752·41-s − 0.413·43-s + 0.799·45-s + 0.904·47-s + 0.142·49-s + 0.191·53-s − 1.47·55-s + 0.346·57-s − 1.20·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8092 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8092 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.422279355\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.422279355\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 0.784T + 3T^{2} \) |
| 5 | \( 1 + 2.25T + 5T^{2} \) |
| 11 | \( 1 - 4.87T + 11T^{2} \) |
| 13 | \( 1 + 0.955T + 13T^{2} \) |
| 19 | \( 1 - 3.33T + 19T^{2} \) |
| 23 | \( 1 + 7.47T + 23T^{2} \) |
| 29 | \( 1 - 0.345T + 29T^{2} \) |
| 31 | \( 1 - 2.60T + 31T^{2} \) |
| 37 | \( 1 - 2.02T + 37T^{2} \) |
| 41 | \( 1 + 4.82T + 41T^{2} \) |
| 43 | \( 1 + 2.71T + 43T^{2} \) |
| 47 | \( 1 - 6.20T + 47T^{2} \) |
| 53 | \( 1 - 1.39T + 53T^{2} \) |
| 59 | \( 1 + 9.23T + 59T^{2} \) |
| 61 | \( 1 - 0.158T + 61T^{2} \) |
| 67 | \( 1 - 1.32T + 67T^{2} \) |
| 71 | \( 1 - 16.2T + 71T^{2} \) |
| 73 | \( 1 - 4.33T + 73T^{2} \) |
| 79 | \( 1 + 4.53T + 79T^{2} \) |
| 83 | \( 1 - 1.58T + 83T^{2} \) |
| 89 | \( 1 + 0.0340T + 89T^{2} \) |
| 97 | \( 1 + 7.27T + 97T^{2} \) |
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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
−7.915692526646267823399556601558, −7.25291682891599682480908869452, −6.44411827281486004994123875725, −5.90174660743413613759005023128, −4.92623940664378513336403147967, −3.95254128236184582132069761017, −3.66966331569954263288393651193, −2.82548047412272707017762659063, −1.82158724925294201897239912465, −0.56527418428145990102978702320,
0.56527418428145990102978702320, 1.82158724925294201897239912465, 2.82548047412272707017762659063, 3.66966331569954263288393651193, 3.95254128236184582132069761017, 4.92623940664378513336403147967, 5.90174660743413613759005023128, 6.44411827281486004994123875725, 7.25291682891599682480908869452, 7.915692526646267823399556601558
