| L(s) = 1 | + 3-s + 2.53·5-s − 7-s + 9-s − 4.91·11-s − 0.165·13-s + 2.53·15-s + 6.84·17-s − 19-s − 21-s − 2.16·23-s + 1.43·25-s + 27-s + 4.70·29-s + 7.07·31-s − 4.91·33-s − 2.53·35-s − 0.746·37-s − 0.165·39-s + 3.07·41-s + 2.53·45-s − 3.88·47-s + 49-s + 6.84·51-s − 0.702·53-s − 12.4·55-s − 57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.13·5-s − 0.377·7-s + 0.333·9-s − 1.48·11-s − 0.0459·13-s + 0.655·15-s + 1.66·17-s − 0.229·19-s − 0.218·21-s − 0.451·23-s + 0.287·25-s + 0.192·27-s + 0.873·29-s + 1.27·31-s − 0.855·33-s − 0.428·35-s − 0.122·37-s − 0.0265·39-s + 0.480·41-s + 0.378·45-s − 0.566·47-s + 0.142·49-s + 0.958·51-s − 0.0964·53-s − 1.68·55-s − 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.938617086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.938617086\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 2.53T + 5T^{2} \) |
| 11 | \( 1 + 4.91T + 11T^{2} \) |
| 13 | \( 1 + 0.165T + 13T^{2} \) |
| 17 | \( 1 - 6.84T + 17T^{2} \) |
| 23 | \( 1 + 2.16T + 23T^{2} \) |
| 29 | \( 1 - 4.70T + 29T^{2} \) |
| 31 | \( 1 - 7.07T + 31T^{2} \) |
| 37 | \( 1 + 0.746T + 37T^{2} \) |
| 41 | \( 1 - 3.07T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 3.88T + 47T^{2} \) |
| 53 | \( 1 + 0.702T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 13.0T + 67T^{2} \) |
| 71 | \( 1 - 8.26T + 71T^{2} \) |
| 73 | \( 1 + 5.05T + 73T^{2} \) |
| 79 | \( 1 + 9.22T + 79T^{2} \) |
| 83 | \( 1 + 8.19T + 83T^{2} \) |
| 89 | \( 1 + 9.38T + 89T^{2} \) |
| 97 | \( 1 + 3.65T + 97T^{2} \) |
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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
−8.199071308110761313342784697226, −7.38538836648791586604998238324, −6.61294482107523560187200836606, −5.76362800162700388333582052164, −5.36934988311884844166887543000, −4.45484557593336648659135364621, −3.36370983777703919817992387521, −2.70120174418543759265743793583, −2.04926760577473412719235540145, −0.866995893451571870112079329990,
0.866995893451571870112079329990, 2.04926760577473412719235540145, 2.70120174418543759265743793583, 3.36370983777703919817992387521, 4.45484557593336648659135364621, 5.36934988311884844166887543000, 5.76362800162700388333582052164, 6.61294482107523560187200836606, 7.38538836648791586604998238324, 8.199071308110761313342784697226
