| L(s) = 1 | − 9.77·3-s − 24.1·5-s + 64.5·7-s − 147.·9-s − 58.5·11-s + 875.·13-s + 235.·15-s + 2.05e3·17-s − 361·19-s − 630.·21-s + 4.90e3·23-s − 2.54e3·25-s + 3.81e3·27-s + 1.70e3·29-s + 3.65e3·31-s + 572.·33-s − 1.55e3·35-s − 2.20e3·37-s − 8.55e3·39-s − 1.22e4·41-s + 1.09e4·43-s + 3.56e3·45-s + 1.21e4·47-s − 1.26e4·49-s − 2.00e4·51-s − 9.12e3·53-s + 1.41e3·55-s + ⋯ |
| L(s) = 1 | − 0.626·3-s − 0.431·5-s + 0.498·7-s − 0.607·9-s − 0.146·11-s + 1.43·13-s + 0.270·15-s + 1.72·17-s − 0.229·19-s − 0.312·21-s + 1.93·23-s − 0.813·25-s + 1.00·27-s + 0.376·29-s + 0.682·31-s + 0.0915·33-s − 0.215·35-s − 0.264·37-s − 0.900·39-s − 1.14·41-s + 0.900·43-s + 0.262·45-s + 0.799·47-s − 0.751·49-s − 1.07·51-s − 0.446·53-s + 0.0630·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.382736258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.382736258\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + 361T \) |
| good | 3 | \( 1 + 9.77T + 243T^{2} \) |
| 5 | \( 1 + 24.1T + 3.12e3T^{2} \) |
| 7 | \( 1 - 64.5T + 1.68e4T^{2} \) |
| 11 | \( 1 + 58.5T + 1.61e5T^{2} \) |
| 13 | \( 1 - 875.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.05e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 4.90e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.70e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.65e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.20e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.22e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.09e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.21e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.12e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.28e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.31e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.29e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.26e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.05e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.74e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.33e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.25e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.16e5T + 8.58e9T^{2} \) |
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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
−13.53397523702197376272851810613, −12.19084503281156914079856152190, −11.35277163568279244322129940486, −10.49669785411483836857834461260, −8.824502654686296245969860278778, −7.77926596899161802417837212272, −6.20547181813379291535045415702, −5.06819908938554341504096715126, −3.34583163915862463514125538093, −0.983451502948084401135874345507,
0.983451502948084401135874345507, 3.34583163915862463514125538093, 5.06819908938554341504096715126, 6.20547181813379291535045415702, 7.77926596899161802417837212272, 8.824502654686296245969860278778, 10.49669785411483836857834461260, 11.35277163568279244322129940486, 12.19084503281156914079856152190, 13.53397523702197376272851810613
