| L(s) = 1 | − 4.59·2-s + 13.1·4-s − 11.7·5-s + 4.20·7-s − 23.4·8-s + 53.8·10-s + 65.7·11-s + 41.4·13-s − 19.3·14-s + 3.05·16-s + 33.7·17-s − 31.8·19-s − 153.·20-s − 302.·22-s − 46.7·23-s + 12.2·25-s − 190.·26-s + 55.1·28-s + 58.6·29-s + 233.·31-s + 173.·32-s − 155.·34-s − 49.3·35-s − 271.·37-s + 146.·38-s + 275.·40-s − 234.·41-s + ⋯ |
| L(s) = 1 | − 1.62·2-s + 1.63·4-s − 1.04·5-s + 0.227·7-s − 1.03·8-s + 1.70·10-s + 1.80·11-s + 0.883·13-s − 0.369·14-s + 0.0477·16-s + 0.482·17-s − 0.384·19-s − 1.71·20-s − 2.92·22-s − 0.423·23-s + 0.0977·25-s − 1.43·26-s + 0.372·28-s + 0.375·29-s + 1.35·31-s + 0.960·32-s − 0.783·34-s − 0.238·35-s − 1.20·37-s + 0.624·38-s + 1.08·40-s − 0.894·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7947813058\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7947813058\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 67 | \( 1 - 67T \) |
| good | 2 | \( 1 + 4.59T + 8T^{2} \) |
| 5 | \( 1 + 11.7T + 125T^{2} \) |
| 7 | \( 1 - 4.20T + 343T^{2} \) |
| 11 | \( 1 - 65.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 41.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 33.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 31.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 46.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 58.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 233.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 271.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 234.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 342.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 280.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 607.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 298.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 541.T + 2.26e5T^{2} \) |
| 71 | \( 1 - 533.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 114.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 164.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 177.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 190.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.60e3T + 9.12e5T^{2} \) |
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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
−10.14357997335451953050650506349, −9.207245991427085590000076228923, −8.503403464755066475372568217537, −7.943026029830583731002538421724, −6.93307189213764103224987754983, −6.23077475692390656262229722476, −4.41591751638188493891931507665, −3.45874006463896523145075579358, −1.69028144112238842776515428480, −0.71861833354593642739779719036,
0.71861833354593642739779719036, 1.69028144112238842776515428480, 3.45874006463896523145075579358, 4.41591751638188493891931507665, 6.23077475692390656262229722476, 6.93307189213764103224987754983, 7.943026029830583731002538421724, 8.503403464755066475372568217537, 9.207245991427085590000076228923, 10.14357997335451953050650506349
