| L(s) = 1 | + 9.91·2-s + 9·3-s + 66.2·4-s − 19.7·5-s + 89.2·6-s + 33.0·7-s + 339.·8-s + 81·9-s − 196.·10-s + 107.·11-s + 596.·12-s − 770.·13-s + 327.·14-s − 178.·15-s + 1.24e3·16-s − 289·17-s + 803.·18-s + 795.·19-s − 1.31e3·20-s + 297.·21-s + 1.06e3·22-s − 2.88e3·23-s + 3.05e3·24-s − 2.73e3·25-s − 7.63e3·26-s + 729·27-s + 2.18e3·28-s + ⋯ |
| L(s) = 1 | + 1.75·2-s + 0.577·3-s + 2.07·4-s − 0.353·5-s + 1.01·6-s + 0.254·7-s + 1.87·8-s + 0.333·9-s − 0.620·10-s + 0.266·11-s + 1.19·12-s − 1.26·13-s + 0.446·14-s − 0.204·15-s + 1.21·16-s − 0.242·17-s + 0.584·18-s + 0.505·19-s − 0.733·20-s + 0.147·21-s + 0.467·22-s − 1.13·23-s + 1.08·24-s − 0.874·25-s − 2.21·26-s + 0.192·27-s + 0.527·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.791770172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.791770172\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 17 | \( 1 + 289T \) |
| good | 2 | \( 1 - 9.91T + 32T^{2} \) |
| 5 | \( 1 + 19.7T + 3.12e3T^{2} \) |
| 7 | \( 1 - 33.0T + 1.68e4T^{2} \) |
| 11 | \( 1 - 107.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 770.T + 3.71e5T^{2} \) |
| 19 | \( 1 - 795.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.88e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.90e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.79e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.37e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.18e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.36e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 9.45e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 5.27e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.42e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.69e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.92e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 8.29e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.58e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.20e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.59e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.96e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.03e5T + 8.58e9T^{2} \) |
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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
−14.31280191105679830639905202493, −13.60921792629898339119974948016, −12.30288214287273168872098124305, −11.63897468789951799680434541388, −9.956485159173852775390355998532, −7.981139712167947136437041696507, −6.67883861863815983133061692059, −5.04677582423571319615957013020, −3.85265863619690549138247951991, −2.36319235306432554411435594820,
2.36319235306432554411435594820, 3.85265863619690549138247951991, 5.04677582423571319615957013020, 6.67883861863815983133061692059, 7.981139712167947136437041696507, 9.956485159173852775390355998532, 11.63897468789951799680434541388, 12.30288214287273168872098124305, 13.60921792629898339119974948016, 14.31280191105679830639905202493
