| L(s) = 1 | + 1.05·2-s − 6.05·3-s − 30.8·4-s − 25·5-s − 6.41·6-s − 213.·7-s − 66.6·8-s − 206.·9-s − 26.4·10-s + 121·11-s + 186.·12-s − 639.·13-s − 226.·14-s + 151.·15-s + 917.·16-s − 1.79e3·17-s − 218.·18-s + 361·19-s + 771.·20-s + 1.29e3·21-s + 128.·22-s + 4.05e3·23-s + 403.·24-s + 625·25-s − 677.·26-s + 2.71e3·27-s + 6.60e3·28-s + ⋯ |
| L(s) = 1 | + 0.187·2-s − 0.388·3-s − 0.964·4-s − 0.447·5-s − 0.0727·6-s − 1.64·7-s − 0.368·8-s − 0.849·9-s − 0.0837·10-s + 0.301·11-s + 0.374·12-s − 1.04·13-s − 0.309·14-s + 0.173·15-s + 0.895·16-s − 1.50·17-s − 0.159·18-s + 0.229·19-s + 0.431·20-s + 0.640·21-s + 0.0564·22-s + 1.59·23-s + 0.142·24-s + 0.200·25-s − 0.196·26-s + 0.717·27-s + 1.59·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 - 1.05T + 32T^{2} \) |
| 3 | \( 1 + 6.05T + 243T^{2} \) |
| 7 | \( 1 + 213.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 639.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.79e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 4.05e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.09e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.26e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.89e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.56e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 5.76e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 903.T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.56e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.16e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.64e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.03e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.80e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.57e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.60e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.22e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.09e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.26e4T + 8.58e9T^{2} \) |
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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
−8.975098711776449058583103993493, −8.090395631440034391195025993465, −6.73232056896720464144665097746, −6.40985701715479038643143797742, −5.12920185549865673889785517962, −4.55256650525421263208166897450, −3.33264547237351384496161505332, −2.76051041168687322944479125711, −0.67977732182244877222240276047, 0,
0.67977732182244877222240276047, 2.76051041168687322944479125711, 3.33264547237351384496161505332, 4.55256650525421263208166897450, 5.12920185549865673889785517962, 6.40985701715479038643143797742, 6.73232056896720464144665097746, 8.090395631440034391195025993465, 8.975098711776449058583103993493
