| L(s) = 1 | + 2.43·3-s − 3.80·5-s − 4.24·7-s + 2.94·9-s + 0.641·11-s − 6.42·13-s − 9.28·15-s + 3.87·17-s − 4.45·19-s − 10.3·21-s + 8.73·23-s + 9.49·25-s − 0.137·27-s + 6.79·29-s − 8.39·31-s + 1.56·33-s + 16.1·35-s + 0.0277·37-s − 15.6·39-s + 5.21·41-s − 2.29·43-s − 11.2·45-s + 10.7·47-s + 11.0·49-s + 9.45·51-s − 10.4·53-s − 2.44·55-s + ⋯ |
| L(s) = 1 | + 1.40·3-s − 1.70·5-s − 1.60·7-s + 0.981·9-s + 0.193·11-s − 1.78·13-s − 2.39·15-s + 0.940·17-s − 1.02·19-s − 2.25·21-s + 1.82·23-s + 1.89·25-s − 0.0264·27-s + 1.26·29-s − 1.50·31-s + 0.272·33-s + 2.73·35-s + 0.00456·37-s − 2.50·39-s + 0.814·41-s − 0.349·43-s − 1.67·45-s + 1.56·47-s + 1.57·49-s + 1.32·51-s − 1.43·53-s − 0.329·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.293504697\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.293504697\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 - 2.43T + 3T^{2} \) |
| 5 | \( 1 + 3.80T + 5T^{2} \) |
| 7 | \( 1 + 4.24T + 7T^{2} \) |
| 11 | \( 1 - 0.641T + 11T^{2} \) |
| 13 | \( 1 + 6.42T + 13T^{2} \) |
| 17 | \( 1 - 3.87T + 17T^{2} \) |
| 19 | \( 1 + 4.45T + 19T^{2} \) |
| 23 | \( 1 - 8.73T + 23T^{2} \) |
| 29 | \( 1 - 6.79T + 29T^{2} \) |
| 31 | \( 1 + 8.39T + 31T^{2} \) |
| 37 | \( 1 - 0.0277T + 37T^{2} \) |
| 41 | \( 1 - 5.21T + 41T^{2} \) |
| 43 | \( 1 + 2.29T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 - 2.18T + 59T^{2} \) |
| 61 | \( 1 - 6.47T + 61T^{2} \) |
| 67 | \( 1 - 1.11T + 67T^{2} \) |
| 71 | \( 1 - 16.1T + 71T^{2} \) |
| 73 | \( 1 - 5.02T + 73T^{2} \) |
| 79 | \( 1 - 17.4T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 - 5.47T + 89T^{2} \) |
| 97 | \( 1 + 17.9T + 97T^{2} \) |
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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
−8.350680767894584637121510438220, −7.77799920226242420862356534686, −7.09192330541249293881774565608, −6.73898840066269857179575767653, −5.29112532756570112426700617274, −4.32193542951768381624997865395, −3.61933332816439720032102750905, −3.05937510593802777695346256014, −2.46311052378855433621965016896, −0.57349471093427629845323336445,
0.57349471093427629845323336445, 2.46311052378855433621965016896, 3.05937510593802777695346256014, 3.61933332816439720032102750905, 4.32193542951768381624997865395, 5.29112532756570112426700617274, 6.73898840066269857179575767653, 7.09192330541249293881774565608, 7.77799920226242420862356534686, 8.350680767894584637121510438220
