| L(s) = 1 | + 2.70·2-s − 9·3-s − 24.7·4-s − 24.3·6-s + 73.4·7-s − 153.·8-s + 81·9-s + 121·11-s + 222.·12-s − 1.05e3·13-s + 198.·14-s + 376.·16-s − 1.85e3·17-s + 218.·18-s + 1.75e3·19-s − 661.·21-s + 326.·22-s − 572.·23-s + 1.37e3·24-s − 2.85e3·26-s − 729·27-s − 1.81e3·28-s + 2.44e3·29-s − 6.03e3·31-s + 5.91e3·32-s − 1.08e3·33-s − 5.00e3·34-s + ⋯ |
| L(s) = 1 | + 0.477·2-s − 0.577·3-s − 0.772·4-s − 0.275·6-s + 0.566·7-s − 0.846·8-s + 0.333·9-s + 0.301·11-s + 0.445·12-s − 1.73·13-s + 0.270·14-s + 0.368·16-s − 1.55·17-s + 0.159·18-s + 1.11·19-s − 0.327·21-s + 0.143·22-s − 0.225·23-s + 0.488·24-s − 0.829·26-s − 0.192·27-s − 0.437·28-s + 0.540·29-s − 1.12·31-s + 1.02·32-s − 0.174·33-s − 0.742·34-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
| \(L(3)\) |
\(\approx\) |
\(0.9339235767\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9339235767\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 2.70T + 32T^{2} \) |
| 7 | \( 1 - 73.4T + 1.68e4T^{2} \) |
| 13 | \( 1 + 1.05e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.85e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.75e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 572.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.44e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.03e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.85e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.27e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.87e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.52e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 397.T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.48e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.68e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.94e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.73e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.25e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.73e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.93e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.49e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.62e4T + 8.58e9T^{2} \) |
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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
−9.524029904354601210790729357877, −8.741228096616533776390021008250, −7.67041214877111714746592378313, −6.81892839193060296015792190885, −5.73528084009427127061722085679, −4.83529050406697233414349209036, −4.48801988709301754235748854540, −3.16434221101772473507993529858, −1.85891838698969048187302143780, −0.41341405238948625160809312714,
0.41341405238948625160809312714, 1.85891838698969048187302143780, 3.16434221101772473507993529858, 4.48801988709301754235748854540, 4.83529050406697233414349209036, 5.73528084009427127061722085679, 6.81892839193060296015792190885, 7.67041214877111714746592378313, 8.741228096616533776390021008250, 9.524029904354601210790729357877
