| L(s) = 1 | + 3.62·2-s + 9·3-s − 18.8·4-s + 57.7·5-s + 32.6·6-s + 251.·7-s − 184.·8-s + 81·9-s + 209.·10-s − 121·11-s − 169.·12-s − 277.·13-s + 910.·14-s + 519.·15-s − 66.1·16-s + 704.·17-s + 293.·18-s − 2.86e3·19-s − 1.08e3·20-s + 2.25e3·21-s − 438.·22-s − 1.06e3·23-s − 1.65e3·24-s + 206.·25-s − 1.00e3·26-s + 729·27-s − 4.73e3·28-s + ⋯ |
| L(s) = 1 | + 0.641·2-s + 0.577·3-s − 0.588·4-s + 1.03·5-s + 0.370·6-s + 1.93·7-s − 1.01·8-s + 0.333·9-s + 0.662·10-s − 0.301·11-s − 0.339·12-s − 0.455·13-s + 1.24·14-s + 0.596·15-s − 0.0646·16-s + 0.591·17-s + 0.213·18-s − 1.81·19-s − 0.607·20-s + 1.11·21-s − 0.193·22-s − 0.420·23-s − 0.588·24-s + 0.0662·25-s − 0.292·26-s + 0.192·27-s − 1.14·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.620263690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.620263690\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 - 3.62T + 32T^{2} \) |
| 5 | \( 1 - 57.7T + 3.12e3T^{2} \) |
| 7 | \( 1 - 251.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 277.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 704.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.86e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.06e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.93e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 644.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.04e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.82e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.05e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.07e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.64e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.29e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 6.83e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.67e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.18e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.39e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.43e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.02e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.95e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.22e4T + 8.58e9T^{2} \) |
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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
−14.99325194156697193416206247668, −14.40524554217815183259477101427, −13.57594132238357617373735500931, −12.34007259110501297233343280900, −10.62439587140345802608846275568, −9.125902147853088621455115513635, −7.959609500874767055202244853026, −5.63390027193040633421602946195, −4.40693769843982712037763401810, −2.03685541957281458688932943194,
2.03685541957281458688932943194, 4.40693769843982712037763401810, 5.63390027193040633421602946195, 7.959609500874767055202244853026, 9.125902147853088621455115513635, 10.62439587140345802608846275568, 12.34007259110501297233343280900, 13.57594132238357617373735500931, 14.40524554217815183259477101427, 14.99325194156697193416206247668
