| L(s) = 1 | − 8.02·2-s − 0.934·3-s + 32.4·4-s − 106.·5-s + 7.50·6-s − 253.·7-s − 3.80·8-s − 242.·9-s + 857.·10-s − 121·11-s − 30.3·12-s − 419.·13-s + 2.03e3·14-s + 99.7·15-s − 1.00e3·16-s − 132.·17-s + 1.94e3·18-s − 2.85e3·19-s − 3.46e3·20-s + 236.·21-s + 971.·22-s + 3.44e3·23-s + 3.55·24-s + 8.27e3·25-s + 3.36e3·26-s + 453.·27-s − 8.23e3·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.0599·3-s + 1.01·4-s − 1.90·5-s + 0.0850·6-s − 1.95·7-s − 0.0210·8-s − 0.996·9-s + 2.71·10-s − 0.301·11-s − 0.0608·12-s − 0.687·13-s + 2.77·14-s + 0.114·15-s − 0.984·16-s − 0.111·17-s + 1.41·18-s − 1.81·19-s − 1.93·20-s + 0.117·21-s + 0.427·22-s + 1.35·23-s + 0.00126·24-s + 2.64·25-s + 0.976·26-s + 0.119·27-s − 1.98·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 583 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 583 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + 121T \) |
| 53 | \( 1 + 2.80e3T \) |
| good | 2 | \( 1 + 8.02T + 32T^{2} \) |
| 3 | \( 1 + 0.934T + 243T^{2} \) |
| 5 | \( 1 + 106.T + 3.12e3T^{2} \) |
| 7 | \( 1 + 253.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 419.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 132.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.85e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.44e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.73e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.28e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.53e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.65e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.92e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.34e4T + 2.29e8T^{2} \) |
| 59 | \( 1 + 2.39e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.93e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.98e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.52e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.11e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.39e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.01e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.69e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.13e5T + 8.58e9T^{2} \) |
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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
−9.171313607283651943125366620423, −8.779665102535647884248311057262, −7.83209174881473027990150438386, −7.09430978244197789175651634331, −6.34273498161203123730861201952, −4.64705231400297385610728825716, −3.48484306123047061945855571713, −2.63277025740983356078224696298, −0.48511524684560820929880825206, 0,
0.48511524684560820929880825206, 2.63277025740983356078224696298, 3.48484306123047061945855571713, 4.64705231400297385610728825716, 6.34273498161203123730861201952, 7.09430978244197789175651634331, 7.83209174881473027990150438386, 8.779665102535647884248311057262, 9.171313607283651943125366620423
