| L(s) = 1 | + 0.780·2-s − 4.65·3-s − 31.3·4-s + 25·5-s − 3.62·6-s − 49.4·8-s − 221.·9-s + 19.5·10-s + 379.·11-s + 145.·12-s + 801.·13-s − 116.·15-s + 965.·16-s + 1.71e3·17-s − 172.·18-s − 2.88e3·19-s − 784.·20-s + 295.·22-s − 3.55e3·23-s + 230.·24-s + 625·25-s + 625.·26-s + 2.15e3·27-s + 504.·29-s − 90.7·30-s − 6.36e3·31-s + 2.33e3·32-s + ⋯ |
| L(s) = 1 | + 0.137·2-s − 0.298·3-s − 0.980·4-s + 0.447·5-s − 0.0411·6-s − 0.273·8-s − 0.910·9-s + 0.0616·10-s + 0.945·11-s + 0.292·12-s + 1.31·13-s − 0.133·15-s + 0.943·16-s + 1.43·17-s − 0.125·18-s − 1.83·19-s − 0.438·20-s + 0.130·22-s − 1.40·23-s + 0.0815·24-s + 0.200·25-s + 0.181·26-s + 0.570·27-s + 0.111·29-s − 0.0184·30-s − 1.18·31-s + 0.403·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 0.780T + 32T^{2} \) |
| 3 | \( 1 + 4.65T + 243T^{2} \) |
| 11 | \( 1 - 379.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 801.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.71e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.88e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.55e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 504.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.36e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.43e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.90e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.49e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.15e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.53e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.64e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.39e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.87e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.64e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.46e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.83e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.14e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.43e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.49e3T + 8.58e9T^{2} \) |
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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
−10.73399572145151201091947223320, −9.726570359574645900923073176062, −8.775679674402029271961784558592, −8.100792362708559963449802377888, −6.21286950414021347012071625871, −5.80675695643749358970443043285, −4.37991097827077823417044003473, −3.36638679847157436232689306841, −1.46092170243864983795257170362, 0,
1.46092170243864983795257170362, 3.36638679847157436232689306841, 4.37991097827077823417044003473, 5.80675695643749358970443043285, 6.21286950414021347012071625871, 8.100792362708559963449802377888, 8.775679674402029271961784558592, 9.726570359574645900923073176062, 10.73399572145151201091947223320
