| L(s) = 1 | − 8.44·2-s + 9·3-s + 39.3·4-s − 36·5-s − 76.0·6-s − 61.9·8-s + 81·9-s + 304.·10-s + 295.·11-s + 354.·12-s − 1.14e3·13-s − 324·15-s − 735.·16-s + 1.03e3·17-s − 684.·18-s + 2.10e3·19-s − 1.41e3·20-s − 2.49e3·22-s − 640.·23-s − 557.·24-s − 1.82e3·25-s + 9.69e3·26-s + 729·27-s + 7.63e3·29-s + 2.73e3·30-s − 966.·31-s + 8.19e3·32-s + ⋯ |
| L(s) = 1 | − 1.49·2-s + 0.577·3-s + 1.22·4-s − 0.643·5-s − 0.862·6-s − 0.342·8-s + 0.333·9-s + 0.961·10-s + 0.736·11-s + 0.709·12-s − 1.88·13-s − 0.371·15-s − 0.718·16-s + 0.866·17-s − 0.497·18-s + 1.33·19-s − 0.791·20-s − 1.09·22-s − 0.252·23-s − 0.197·24-s − 0.585·25-s + 2.81·26-s + 0.192·27-s + 1.68·29-s + 0.555·30-s − 0.180·31-s + 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 8.44T + 32T^{2} \) |
| 5 | \( 1 + 36T + 3.12e3T^{2} \) |
| 11 | \( 1 - 295.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.14e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.03e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.10e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 640.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.63e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 966.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.77e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.19e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.98e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.79e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 7.11e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.08e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.38e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.46e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.84e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.52e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.30e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.03e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.60e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.53e5T + 8.58e9T^{2} \) |
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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
−11.61251590380328950103372197288, −9.970422729148389464513290097583, −9.739936415910330830400651625393, −8.442685175488656226813295807271, −7.68357269320336289447196818363, −6.89135113873450563609443308655, −4.78747278162094980288634768130, −3.10496716988514766309135713734, −1.50548573645969688894219662078, 0,
1.50548573645969688894219662078, 3.10496716988514766309135713734, 4.78747278162094980288634768130, 6.89135113873450563609443308655, 7.68357269320336289447196818363, 8.442685175488656226813295807271, 9.739936415910330830400651625393, 9.970422729148389464513290097583, 11.61251590380328950103372197288
