| L(s) = 1 | + 5.93·2-s + 9·3-s + 3.25·4-s + 53.4·6-s + 105.·7-s − 170.·8-s + 81·9-s − 121·11-s + 29.2·12-s − 124.·13-s + 626.·14-s − 1.11e3·16-s − 1.76e3·17-s + 480.·18-s + 1.99e3·19-s + 949.·21-s − 718.·22-s + 2.54e3·23-s − 1.53e3·24-s − 737.·26-s + 729·27-s + 343.·28-s − 320.·29-s + 8.55e3·31-s − 1.17e3·32-s − 1.08e3·33-s − 1.04e4·34-s + ⋯ |
| L(s) = 1 | + 1.04·2-s + 0.577·3-s + 0.101·4-s + 0.605·6-s + 0.814·7-s − 0.942·8-s + 0.333·9-s − 0.301·11-s + 0.0586·12-s − 0.203·13-s + 0.854·14-s − 1.09·16-s − 1.48·17-s + 0.349·18-s + 1.27·19-s + 0.470·21-s − 0.316·22-s + 1.00·23-s − 0.544·24-s − 0.214·26-s + 0.192·27-s + 0.0827·28-s − 0.0707·29-s + 1.59·31-s − 0.202·32-s − 0.174·33-s − 1.55·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.672101435\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.672101435\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 - 5.93T + 32T^{2} \) |
| 7 | \( 1 - 105.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 124.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.76e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.99e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.54e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 320.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.55e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.12e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 4.53e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.34e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.41e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.84e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.82e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.69e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.37e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.60e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.61e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.18e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.51e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.83e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.17e4T + 8.58e9T^{2} \) |
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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
−9.235984762156368444439370894934, −8.742916753545834704461532741803, −7.71618667942349913720990982547, −6.83886158737160248378471625185, −5.70484811032339513554123393450, −4.81759790502416662146549863575, −4.25010561627378716693411724898, −3.07697341493720900708936047216, −2.28867130242527703407877818862, −0.818952863501770882379823787778,
0.818952863501770882379823787778, 2.28867130242527703407877818862, 3.07697341493720900708936047216, 4.25010561627378716693411724898, 4.81759790502416662146549863575, 5.70484811032339513554123393450, 6.83886158737160248378471625185, 7.71618667942349913720990982547, 8.742916753545834704461532741803, 9.235984762156368444439370894934
