| L(s) = 1 | − 4.30·3-s − 28.3·7-s − 8.49·9-s + 65.2·11-s − 33.6·13-s + 73.3·17-s + 134.·19-s + 121.·21-s + 14.7·23-s + 152.·27-s − 224.·29-s + 68.8·31-s − 280.·33-s − 196.·37-s + 144.·39-s − 143.·41-s − 15.0·43-s − 134.·47-s + 458.·49-s − 315.·51-s + 262.·53-s − 576.·57-s − 119.·59-s + 16.5·61-s + 240.·63-s − 545.·67-s − 63.2·69-s + ⋯ |
| L(s) = 1 | − 0.827·3-s − 1.52·7-s − 0.314·9-s + 1.78·11-s − 0.718·13-s + 1.04·17-s + 1.61·19-s + 1.26·21-s + 0.133·23-s + 1.08·27-s − 1.43·29-s + 0.398·31-s − 1.48·33-s − 0.872·37-s + 0.594·39-s − 0.545·41-s − 0.0534·43-s − 0.417·47-s + 1.33·49-s − 0.865·51-s + 0.681·53-s − 1.34·57-s − 0.264·59-s + 0.0347·61-s + 0.481·63-s − 0.994·67-s − 0.110·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 4.30T + 27T^{2} \) |
| 7 | \( 1 + 28.3T + 343T^{2} \) |
| 11 | \( 1 - 65.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 33.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 73.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 134.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 14.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 224.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 68.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 196.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 143.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 15.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + 134.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 262.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 119.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 16.5T + 2.26e5T^{2} \) |
| 67 | \( 1 + 545.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 199.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 43.2T + 3.89e5T^{2} \) |
| 79 | \( 1 - 438.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.22e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 723.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.13e3T + 9.12e5T^{2} \) |
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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
−9.581711052641439935003840835169, −8.887682436360886198220087643716, −7.42764967466285264250247319530, −6.72332419712555598769501689796, −5.95017205605691022085007988502, −5.20671458341954719046144052941, −3.76684670699694119916689970973, −3.04401721947458475719243641990, −1.20648376297121202906859645087, 0,
1.20648376297121202906859645087, 3.04401721947458475719243641990, 3.76684670699694119916689970973, 5.20671458341954719046144052941, 5.95017205605691022085007988502, 6.72332419712555598769501689796, 7.42764967466285264250247319530, 8.887682436360886198220087643716, 9.581711052641439935003840835169
