| L(s) = 1 | + 7.34·2-s + 22·4-s + 58.7·5-s + 167·7-s − 73.4·8-s + 432·10-s − 764.·11-s − 235·13-s + 1.22e3·14-s − 1.24e3·16-s + 176.·17-s + 1.36e3·19-s + 1.29e3·20-s − 5.61e3·22-s − 2.41e3·23-s + 331·25-s − 1.72e3·26-s + 3.67e3·28-s + 470.·29-s + 3.50e3·31-s − 6.78e3·32-s + 1.29e3·34-s + 9.81e3·35-s + 1.31e4·37-s + 1.00e4·38-s − 4.32e3·40-s + 9.40e3·41-s + ⋯ |
| L(s) = 1 | + 1.29·2-s + 0.687·4-s + 1.05·5-s + 1.28·7-s − 0.405·8-s + 1.36·10-s − 1.90·11-s − 0.385·13-s + 1.67·14-s − 1.21·16-s + 0.148·17-s + 0.864·19-s + 0.722·20-s − 2.47·22-s − 0.950·23-s + 0.105·25-s − 0.500·26-s + 0.885·28-s + 0.103·29-s + 0.654·31-s − 1.17·32-s + 0.192·34-s + 1.35·35-s + 1.57·37-s + 1.12·38-s − 0.426·40-s + 0.873·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.953267688\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.953267688\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 7.34T + 32T^{2} \) |
| 5 | \( 1 - 58.7T + 3.12e3T^{2} \) |
| 7 | \( 1 - 167T + 1.68e4T^{2} \) |
| 11 | \( 1 + 764.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 235T + 3.71e5T^{2} \) |
| 17 | \( 1 - 176.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.36e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.41e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 470.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.50e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.31e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 9.40e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 104T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.05e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.05e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.07e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 7.39e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.88e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.46e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.46e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.29e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.32e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.97e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.96e4T + 8.58e9T^{2} \) |
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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
−15.91368349828083401634052292965, −14.66070449210053976603178890432, −13.80117634488285171658816933005, −12.89415706528061783232935454164, −11.45355905347144470119494466548, −9.935885119629252332146229310373, −7.930641282386248022984232602830, −5.74527203674604307646717345609, −4.83916291033150074860993440614, −2.44692543196522632817817329616,
2.44692543196522632817817329616, 4.83916291033150074860993440614, 5.74527203674604307646717345609, 7.930641282386248022984232602830, 9.935885119629252332146229310373, 11.45355905347144470119494466548, 12.89415706528061783232935454164, 13.80117634488285171658816933005, 14.66070449210053976603178890432, 15.91368349828083401634052292965
