| L(s) = 1 | − 0.312·2-s − 15.4·3-s − 31.9·4-s − 25·5-s + 4.83·6-s − 30.9·7-s + 19.9·8-s − 3.17·9-s + 7.80·10-s − 121·11-s + 494.·12-s − 775.·13-s + 9.67·14-s + 387.·15-s + 1.01e3·16-s − 666.·17-s + 0.990·18-s − 361·19-s + 797.·20-s + 479.·21-s + 37.7·22-s − 4.19e3·23-s − 309.·24-s + 625·25-s + 242.·26-s + 3.81e3·27-s + 988.·28-s + ⋯ |
| L(s) = 1 | − 0.0552·2-s − 0.993·3-s − 0.996·4-s − 0.447·5-s + 0.0548·6-s − 0.238·7-s + 0.110·8-s − 0.0130·9-s + 0.0246·10-s − 0.301·11-s + 0.990·12-s − 1.27·13-s + 0.0131·14-s + 0.444·15-s + 0.990·16-s − 0.559·17-s + 0.000720·18-s − 0.229·19-s + 0.445·20-s + 0.237·21-s + 0.0166·22-s − 1.65·23-s − 0.109·24-s + 0.200·25-s + 0.0702·26-s + 1.00·27-s + 0.238·28-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 | \( 1 + 0.312T + 32T^{2} \) |
| 3 | \( 1 + 15.4T + 243T^{2} \) |
| 7 | \( 1 + 30.9T + 1.68e4T^{2} \) |
| 13 | \( 1 + 775.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 666.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 4.19e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.78e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.53e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.01e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.87e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.21e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.50e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 8.10e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.94e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.76e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.44e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.09e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.19e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.57e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.73e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.52e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.21e4T + 8.58e9T^{2} \) |
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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
−8.661788287944469802062124501195, −8.093332810193230331959490986991, −7.02107168635067103682142705495, −6.12337106789344063364296404423, −5.17008874960058885494493596974, −4.63643200239401264101487992984, −3.63394499257998387845611177939, −2.32323570703160490193974617341, −0.65298507157357856875307505743, 0,
0.65298507157357856875307505743, 2.32323570703160490193974617341, 3.63394499257998387845611177939, 4.63643200239401264101487992984, 5.17008874960058885494493596974, 6.12337106789344063364296404423, 7.02107168635067103682142705495, 8.093332810193230331959490986991, 8.661788287944469802062124501195
