| L(s) = 1 | − 10.5·2-s + 22.4·3-s + 80.2·4-s − 25·5-s − 238.·6-s + 228.·7-s − 511.·8-s + 261.·9-s + 264.·10-s − 121·11-s + 1.80e3·12-s − 327.·13-s − 2.41e3·14-s − 561.·15-s + 2.85e3·16-s − 132.·17-s − 2.77e3·18-s − 361·19-s − 2.00e3·20-s + 5.12e3·21-s + 1.28e3·22-s − 108.·23-s − 1.14e4·24-s + 625·25-s + 3.46e3·26-s + 418.·27-s + 1.83e4·28-s + ⋯ |
| L(s) = 1 | − 1.87·2-s + 1.44·3-s + 2.50·4-s − 0.447·5-s − 2.69·6-s + 1.75·7-s − 2.82·8-s + 1.07·9-s + 0.837·10-s − 0.301·11-s + 3.61·12-s − 0.537·13-s − 3.29·14-s − 0.644·15-s + 2.78·16-s − 0.111·17-s − 2.01·18-s − 0.229·19-s − 1.12·20-s + 2.53·21-s + 0.564·22-s − 0.0427·23-s − 4.07·24-s + 0.200·25-s + 1.00·26-s + 0.110·27-s + 4.41·28-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 | \( 1 + 10.5T + 32T^{2} \) |
| 3 | \( 1 - 22.4T + 243T^{2} \) |
| 7 | \( 1 - 228.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 327.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 132.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 108.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.73e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.93e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.50e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.61e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 9.15e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.23e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.56e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.35e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.96e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.88e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.73e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.27e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.76e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.18e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.48e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.50e4T + 8.58e9T^{2} \) |
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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
−8.592284595385305308194717540113, −8.216818074116761491227236065084, −7.54092909481908968089663946202, −7.06430033288776239132074717346, −5.43065258412131434906076981745, −4.11044001958040015921067739710, −2.82505046191587721820419974228, −2.03550553149828098733277771001, −1.36886064127297674285616119331, 0,
1.36886064127297674285616119331, 2.03550553149828098733277771001, 2.82505046191587721820419974228, 4.11044001958040015921067739710, 5.43065258412131434906076981745, 7.06430033288776239132074717346, 7.54092909481908968089663946202, 8.216818074116761491227236065084, 8.592284595385305308194717540113
