| L(s) = 1 | − 10.3·2-s + 75.1·4-s + 71.3·5-s − 210.·7-s − 447.·8-s − 738.·10-s + 170.·11-s + 605.·13-s + 2.18e3·14-s + 2.22e3·16-s + 1.33e3·17-s − 1.79e3·19-s + 5.36e3·20-s − 1.76e3·22-s + 529·23-s + 1.96e3·25-s − 6.26e3·26-s − 1.58e4·28-s + 3.00e3·29-s − 9.21e3·31-s − 8.71e3·32-s − 1.38e4·34-s − 1.50e4·35-s + 6.02e3·37-s + 1.85e4·38-s − 3.19e4·40-s + 6.60e3·41-s + ⋯ |
| L(s) = 1 | − 1.83·2-s + 2.34·4-s + 1.27·5-s − 1.62·7-s − 2.47·8-s − 2.33·10-s + 0.424·11-s + 0.993·13-s + 2.97·14-s + 2.17·16-s + 1.12·17-s − 1.14·19-s + 2.99·20-s − 0.776·22-s + 0.208·23-s + 0.629·25-s − 1.81·26-s − 3.82·28-s + 0.663·29-s − 1.72·31-s − 1.50·32-s − 2.05·34-s − 2.07·35-s + 0.723·37-s + 2.08·38-s − 3.15·40-s + 0.613·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8792892855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8792892855\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - 529T \) |
| good | 2 | \( 1 + 10.3T + 32T^{2} \) |
| 5 | \( 1 - 71.3T + 3.12e3T^{2} \) |
| 7 | \( 1 + 210.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 170.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 605.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.33e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.79e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 3.00e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.21e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.02e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 6.60e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.29e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.01e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.60e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.71e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.16e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.78e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.32e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.26e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.93e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.14e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.36e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.51e5T + 8.58e9T^{2} \) |
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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
−10.91224734944505751249781912920, −10.26125700951616779189223272311, −9.353806859755846316031085474835, −9.059555032174094431956471262579, −7.63926427850683604275938822656, −6.31557429467283411828420051311, −6.11023259150973100197475096958, −3.29235065361214650931423599834, −1.99691884836966262829612765062, −0.75172921475700371089013856349,
0.75172921475700371089013856349, 1.99691884836966262829612765062, 3.29235065361214650931423599834, 6.11023259150973100197475096958, 6.31557429467283411828420051311, 7.63926427850683604275938822656, 9.059555032174094431956471262579, 9.353806859755846316031085474835, 10.26125700951616779189223272311, 10.91224734944505751249781912920
