| L(s) = 1 | − 9.55·2-s − 7.71·3-s + 59.2·4-s + 25·5-s + 73.7·6-s + 175.·7-s − 260.·8-s − 183.·9-s − 238.·10-s − 121·11-s − 457.·12-s − 84.5·13-s − 1.67e3·14-s − 192.·15-s + 589.·16-s − 736.·17-s + 1.75e3·18-s + 361·19-s + 1.48e3·20-s − 1.35e3·21-s + 1.15e3·22-s + 3.31e3·23-s + 2.00e3·24-s + 625·25-s + 807.·26-s + 3.29e3·27-s + 1.03e4·28-s + ⋯ |
| L(s) = 1 | − 1.68·2-s − 0.495·3-s + 1.85·4-s + 0.447·5-s + 0.836·6-s + 1.35·7-s − 1.43·8-s − 0.754·9-s − 0.755·10-s − 0.301·11-s − 0.916·12-s − 0.138·13-s − 2.28·14-s − 0.221·15-s + 0.575·16-s − 0.618·17-s + 1.27·18-s + 0.229·19-s + 0.827·20-s − 0.670·21-s + 0.509·22-s + 1.30·23-s + 0.711·24-s + 0.200·25-s + 0.234·26-s + 0.868·27-s + 2.50·28-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 | \( 1 + 9.55T + 32T^{2} \) |
| 3 | \( 1 + 7.71T + 243T^{2} \) |
| 7 | \( 1 - 175.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 84.5T + 3.71e5T^{2} \) |
| 17 | \( 1 + 736.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 3.31e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.61e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 256.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.04e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.69e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 7.98e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 5.24e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 7.15e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.77e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.05e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.01e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.78e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.10e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.76e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.46e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.31e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.55e5T + 8.58e9T^{2} \) |
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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
−8.848434552392617528069295279302, −8.108216689518001381066027452764, −7.38148878297149848356628444177, −6.49456823796116702856302387803, −5.46781547374723429271575042546, −4.69444991456693005135623868737, −2.85159652694021744583849814019, −1.90036041268710053730390271576, −1.04045600342646323694842908129, 0,
1.04045600342646323694842908129, 1.90036041268710053730390271576, 2.85159652694021744583849814019, 4.69444991456693005135623868737, 5.46781547374723429271575042546, 6.49456823796116702856302387803, 7.38148878297149848356628444177, 8.108216689518001381066027452764, 8.848434552392617528069295279302
