| L(s) = 1 | − 1.59·2-s + 3·3-s − 5.45·4-s + 8.73·5-s − 4.78·6-s − 29.0·7-s + 21.4·8-s + 9·9-s − 13.9·10-s − 16.3·12-s − 31.1·13-s + 46.2·14-s + 26.2·15-s + 9.42·16-s + 109.·17-s − 14.3·18-s − 81.2·19-s − 47.6·20-s − 87.0·21-s + 143.·23-s + 64.3·24-s − 48.6·25-s + 49.6·26-s + 27·27-s + 158.·28-s + 85.5·29-s − 41.7·30-s + ⋯ |
| L(s) = 1 | − 0.563·2-s + 0.577·3-s − 0.682·4-s + 0.781·5-s − 0.325·6-s − 1.56·7-s + 0.948·8-s + 0.333·9-s − 0.440·10-s − 0.393·12-s − 0.663·13-s + 0.883·14-s + 0.451·15-s + 0.147·16-s + 1.56·17-s − 0.187·18-s − 0.980·19-s − 0.532·20-s − 0.904·21-s + 1.30·23-s + 0.547·24-s − 0.389·25-s + 0.374·26-s + 0.192·27-s + 1.06·28-s + 0.547·29-s − 0.254·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.329382004\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.329382004\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.59T + 8T^{2} \) |
| 5 | \( 1 - 8.73T + 125T^{2} \) |
| 7 | \( 1 + 29.0T + 343T^{2} \) |
| 13 | \( 1 + 31.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 109.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 81.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 143.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 85.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 308.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 120.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 134.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 207.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 328.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 338.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 657.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 414.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 122.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.03e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 772.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 477.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.12e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 472.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.37e3T + 9.12e5T^{2} \) |
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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
−10.24563920507507730427951545209, −10.05463115731257956974379086356, −9.272621154508587239630760570438, −8.470613706924115539863774919257, −7.33839364275015520573928286484, −6.30839304734958352742320078045, −5.10743244778445591163025299232, −3.71920781050341466681657428245, −2.56931066461482979505730390307, −0.825568935356703110791253102644,
0.825568935356703110791253102644, 2.56931066461482979505730390307, 3.71920781050341466681657428245, 5.10743244778445591163025299232, 6.30839304734958352742320078045, 7.33839364275015520573928286484, 8.470613706924115539863774919257, 9.272621154508587239630760570438, 10.05463115731257956974379086356, 10.24563920507507730427951545209
