| L(s) = 1 | − 5.20·3-s + 7.79·5-s + 26.8·7-s + 0.114·9-s + 40.7·11-s + 24.2·13-s − 40.6·15-s + 70.2·17-s − 19·19-s − 139.·21-s + 37.2·23-s − 64.1·25-s + 139.·27-s − 171.·29-s + 64.4·31-s − 212.·33-s + 209.·35-s − 432.·37-s − 126.·39-s + 270.·41-s + 49.8·43-s + 0.889·45-s + 362.·47-s + 378.·49-s − 365.·51-s − 515.·53-s + 318.·55-s + ⋯ |
| L(s) = 1 | − 1.00·3-s + 0.697·5-s + 1.45·7-s + 0.00422·9-s + 1.11·11-s + 0.516·13-s − 0.699·15-s + 1.00·17-s − 0.229·19-s − 1.45·21-s + 0.338·23-s − 0.513·25-s + 0.997·27-s − 1.10·29-s + 0.373·31-s − 1.12·33-s + 1.01·35-s − 1.92·37-s − 0.517·39-s + 1.02·41-s + 0.176·43-s + 0.00294·45-s + 1.12·47-s + 1.10·49-s − 1.00·51-s − 1.33·53-s + 0.779·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.078258311\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.078258311\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 3 | \( 1 + 5.20T + 27T^{2} \) |
| 5 | \( 1 - 7.79T + 125T^{2} \) |
| 7 | \( 1 - 26.8T + 343T^{2} \) |
| 11 | \( 1 - 40.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 24.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 70.2T + 4.91e3T^{2} \) |
| 23 | \( 1 - 37.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 171.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 64.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 432.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 270.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 49.8T + 7.95e4T^{2} \) |
| 47 | \( 1 - 362.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 515.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 71.0T + 2.05e5T^{2} \) |
| 61 | \( 1 - 735.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 224.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 67.6T + 3.57e5T^{2} \) |
| 73 | \( 1 + 89.3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.21e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 948.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 877.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 812.T + 9.12e5T^{2} \) |
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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
−10.50651335642286514536766130423, −9.386372375895951674607288847358, −8.547803983080169260679975264185, −7.53252022773912265922440155527, −6.39297022594691203992160897344, −5.64287088296800008032377407203, −4.94969597144863238709422684616, −3.73564205169121243630964144597, −1.92812759620926424283288195898, −0.977474511156452559523927649636,
0.977474511156452559523927649636, 1.92812759620926424283288195898, 3.73564205169121243630964144597, 4.94969597144863238709422684616, 5.64287088296800008032377407203, 6.39297022594691203992160897344, 7.53252022773912265922440155527, 8.547803983080169260679975264185, 9.386372375895951674607288847358, 10.50651335642286514536766130423
