| L(s) = 1 | + 1.31·3-s + 5-s − 4.66·7-s − 1.28·9-s − 2.23·11-s + 2.80·13-s + 1.31·15-s + 7.63·17-s + 1.36·19-s − 6.11·21-s − 23-s + 25-s − 5.61·27-s − 8.94·29-s − 1.58·31-s − 2.92·33-s − 4.66·35-s + 1.40·37-s + 3.67·39-s + 10.7·41-s − 1.28·45-s − 7.26·47-s + 14.7·49-s + 10.0·51-s + 8.38·53-s − 2.23·55-s + 1.78·57-s + ⋯ |
| L(s) = 1 | + 0.756·3-s + 0.447·5-s − 1.76·7-s − 0.427·9-s − 0.672·11-s + 0.776·13-s + 0.338·15-s + 1.85·17-s + 0.312·19-s − 1.33·21-s − 0.208·23-s + 0.200·25-s − 1.08·27-s − 1.66·29-s − 0.284·31-s − 0.508·33-s − 0.788·35-s + 0.230·37-s + 0.587·39-s + 1.67·41-s − 0.191·45-s − 1.05·47-s + 2.10·49-s + 1.40·51-s + 1.15·53-s − 0.300·55-s + 0.236·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.988514080\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.988514080\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 1.31T + 3T^{2} \) |
| 7 | \( 1 + 4.66T + 7T^{2} \) |
| 11 | \( 1 + 2.23T + 11T^{2} \) |
| 13 | \( 1 - 2.80T + 13T^{2} \) |
| 17 | \( 1 - 7.63T + 17T^{2} \) |
| 19 | \( 1 - 1.36T + 19T^{2} \) |
| 29 | \( 1 + 8.94T + 29T^{2} \) |
| 31 | \( 1 + 1.58T + 31T^{2} \) |
| 37 | \( 1 - 1.40T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 7.26T + 47T^{2} \) |
| 53 | \( 1 - 8.38T + 53T^{2} \) |
| 59 | \( 1 - 4.88T + 59T^{2} \) |
| 61 | \( 1 + 4.33T + 61T^{2} \) |
| 67 | \( 1 + 8.54T + 67T^{2} \) |
| 71 | \( 1 - 8.81T + 71T^{2} \) |
| 73 | \( 1 + 5.26T + 73T^{2} \) |
| 79 | \( 1 - 7.08T + 79T^{2} \) |
| 83 | \( 1 - 4.59T + 83T^{2} \) |
| 89 | \( 1 + 4.70T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−7.78060574004560494772223708308, −7.42956694347856034411510253081, −6.38944868740709038134052813752, −5.75979415676666286952175286846, −5.45338808334148311638054559557, −3.96726903482877909246961814215, −3.32818361101938144276171574346, −2.94839873609169037041256057867, −1.99580842975567050106943801026, −0.65945014658106891208558241282,
0.65945014658106891208558241282, 1.99580842975567050106943801026, 2.94839873609169037041256057867, 3.32818361101938144276171574346, 3.96726903482877909246961814215, 5.45338808334148311638054559557, 5.75979415676666286952175286846, 6.38944868740709038134052813752, 7.42956694347856034411510253081, 7.78060574004560494772223708308
