| L(s) = 1 | − 2.80·3-s + 3.19·5-s + 4.58·7-s + 4.87·9-s − 0.205·11-s − 3.43·13-s − 8.97·15-s − 17-s + 0.881·19-s − 12.8·21-s + 6.90·23-s + 5.22·25-s − 5.26·27-s + 1.28·29-s + 2.36·31-s + 0.576·33-s + 14.6·35-s + 1.95·37-s + 9.63·39-s + 3.63·41-s − 4.79·43-s + 15.5·45-s + 7.86·47-s + 14.0·49-s + 2.80·51-s − 10.7·53-s − 0.657·55-s + ⋯ |
| L(s) = 1 | − 1.62·3-s + 1.43·5-s + 1.73·7-s + 1.62·9-s − 0.0619·11-s − 0.952·13-s − 2.31·15-s − 0.242·17-s + 0.202·19-s − 2.81·21-s + 1.43·23-s + 1.04·25-s − 1.01·27-s + 0.239·29-s + 0.424·31-s + 0.100·33-s + 2.48·35-s + 0.320·37-s + 1.54·39-s + 0.568·41-s − 0.731·43-s + 2.32·45-s + 1.14·47-s + 2.00·49-s + 0.392·51-s − 1.47·53-s − 0.0886·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.846618829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.846618829\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.80T + 3T^{2} \) |
| 5 | \( 1 - 3.19T + 5T^{2} \) |
| 7 | \( 1 - 4.58T + 7T^{2} \) |
| 11 | \( 1 + 0.205T + 11T^{2} \) |
| 13 | \( 1 + 3.43T + 13T^{2} \) |
| 19 | \( 1 - 0.881T + 19T^{2} \) |
| 23 | \( 1 - 6.90T + 23T^{2} \) |
| 29 | \( 1 - 1.28T + 29T^{2} \) |
| 31 | \( 1 - 2.36T + 31T^{2} \) |
| 37 | \( 1 - 1.95T + 37T^{2} \) |
| 41 | \( 1 - 3.63T + 41T^{2} \) |
| 43 | \( 1 + 4.79T + 43T^{2} \) |
| 47 | \( 1 - 7.86T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 61 | \( 1 - 2.10T + 61T^{2} \) |
| 67 | \( 1 - 7.51T + 67T^{2} \) |
| 71 | \( 1 - 4.32T + 71T^{2} \) |
| 73 | \( 1 - 5.29T + 73T^{2} \) |
| 79 | \( 1 + 2.22T + 79T^{2} \) |
| 83 | \( 1 + 3.50T + 83T^{2} \) |
| 89 | \( 1 + 9.35T + 89T^{2} \) |
| 97 | \( 1 - 7.53T + 97T^{2} \) |
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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
−8.443366373852216912428904407091, −7.50561058395458901211076871307, −6.82468238098498749409249293885, −6.08100052279362972763272614617, −5.28698767138598861609198832186, −5.04801980858337104487536744197, −4.39199941403646810264593803695, −2.62260678211485211929988766897, −1.71683039012169790916056596190, −0.912032461477659162657046760908,
0.912032461477659162657046760908, 1.71683039012169790916056596190, 2.62260678211485211929988766897, 4.39199941403646810264593803695, 5.04801980858337104487536744197, 5.28698767138598861609198832186, 6.08100052279362972763272614617, 6.82468238098498749409249293885, 7.50561058395458901211076871307, 8.443366373852216912428904407091
