| L(s) = 1 | + 9·3-s + 46.6·5-s + 58.6·7-s + 81·9-s + 125.·11-s + 351.·13-s + 419.·15-s − 885.·17-s − 361·19-s + 527.·21-s − 936.·23-s − 952.·25-s + 729·27-s + 3.11e3·29-s + 1.05e4·31-s + 1.12e3·33-s + 2.73e3·35-s + 1.06e4·37-s + 3.15e3·39-s + 1.12e4·41-s − 5.84e3·43-s + 3.77e3·45-s + 4.84e3·47-s − 1.33e4·49-s − 7.97e3·51-s − 3.66e4·53-s + 5.85e3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.833·5-s + 0.452·7-s + 0.333·9-s + 0.312·11-s + 0.576·13-s + 0.481·15-s − 0.743·17-s − 0.229·19-s + 0.261·21-s − 0.369·23-s − 0.304·25-s + 0.192·27-s + 0.688·29-s + 1.97·31-s + 0.180·33-s + 0.377·35-s + 1.27·37-s + 0.332·39-s + 1.04·41-s − 0.482·43-s + 0.277·45-s + 0.320·47-s − 0.795·49-s − 0.429·51-s − 1.79·53-s + 0.260·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.050087806\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.050087806\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| 19 | \( 1 + 361T \) |
| good | 5 | \( 1 - 46.6T + 3.12e3T^{2} \) |
| 7 | \( 1 - 58.6T + 1.68e4T^{2} \) |
| 11 | \( 1 - 125.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 351.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 885.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 936.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.11e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.05e4T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.06e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.12e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 5.84e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 4.84e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.66e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 7.92e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.94e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.51e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.90e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.68e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.87e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.18e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.59e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.83e4T + 8.58e9T^{2} \) |
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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
−9.423895697223750507609644049812, −8.452004892881718769802243846529, −7.925902211100438681406401302772, −6.61622487070853669909034336444, −6.09432820353000649001563842661, −4.85063019900995481660586362539, −4.03477325268450676351575400754, −2.76340978845520624548583303013, −1.92594665863398284872737835602, −0.904569796265308037327283552690,
0.904569796265308037327283552690, 1.92594665863398284872737835602, 2.76340978845520624548583303013, 4.03477325268450676351575400754, 4.85063019900995481660586362539, 6.09432820353000649001563842661, 6.61622487070853669909034336444, 7.925902211100438681406401302772, 8.452004892881718769802243846529, 9.423895697223750507609644049812
