| L(s) = 1 | − 7.82·2-s − 16.3·3-s + 29.2·4-s − 25·5-s + 127.·6-s + 125.·7-s + 21.7·8-s + 22.8·9-s + 195.·10-s − 476.·12-s − 532.·13-s − 981.·14-s + 407.·15-s − 1.10e3·16-s + 1.37e3·17-s − 178.·18-s + 554.·19-s − 730.·20-s − 2.04e3·21-s + 4.25e3·23-s − 353.·24-s + 625·25-s + 4.16e3·26-s + 3.58e3·27-s + 3.66e3·28-s + 6.97e3·29-s − 3.18e3·30-s + ⋯ |
| L(s) = 1 | − 1.38·2-s − 1.04·3-s + 0.913·4-s − 0.447·5-s + 1.44·6-s + 0.967·7-s + 0.119·8-s + 0.0939·9-s + 0.618·10-s − 0.955·12-s − 0.873·13-s − 1.33·14-s + 0.467·15-s − 1.07·16-s + 1.15·17-s − 0.129·18-s + 0.352·19-s − 0.408·20-s − 1.01·21-s + 1.67·23-s − 0.125·24-s + 0.200·25-s + 1.20·26-s + 0.947·27-s + 0.883·28-s + 1.53·29-s − 0.647·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.6675664770\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6675664770\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 \) |
| good | 2 | \( 1 + 7.82T + 32T^{2} \) |
| 3 | \( 1 + 16.3T + 243T^{2} \) |
| 7 | \( 1 - 125.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 532.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.37e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 554.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.25e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.97e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.13e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.38e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 679.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.72e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.51e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.54e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.75e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.05e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.87e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.25e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.37e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.86e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.22e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.15e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.90e4T + 8.58e9T^{2} \) |
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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
−10.04360155506071943045215430057, −8.940137634939855211390826685917, −8.140972855757827280293851941489, −7.44552487380953130853365935941, −6.57315337176407659339677751805, −5.19166865581397961277699678419, −4.67678073300255166720507201100, −2.84946332232928649030109290245, −1.28890654916514634472859819360, −0.60205100668672980087913108295,
0.60205100668672980087913108295, 1.28890654916514634472859819360, 2.84946332232928649030109290245, 4.67678073300255166720507201100, 5.19166865581397961277699678419, 6.57315337176407659339677751805, 7.44552487380953130853365935941, 8.140972855757827280293851941489, 8.940137634939855211390826685917, 10.04360155506071943045215430057
