| L(s) = 1 | + 6.46·2-s − 9·3-s + 9.74·4-s − 95.9·5-s − 58.1·6-s + 79.3·7-s − 143.·8-s + 81·9-s − 619.·10-s + 411.·11-s − 87.6·12-s − 318.·13-s + 512.·14-s + 863.·15-s − 1.24e3·16-s + 1.18e3·17-s + 523.·18-s + 1.50e3·19-s − 934.·20-s − 714.·21-s + 2.66e3·22-s − 2.53e3·23-s + 1.29e3·24-s + 6.07e3·25-s − 2.05e3·26-s − 729·27-s + 773.·28-s + ⋯ |
| L(s) = 1 | + 1.14·2-s − 0.577·3-s + 0.304·4-s − 1.71·5-s − 0.659·6-s + 0.612·7-s − 0.794·8-s + 0.333·9-s − 1.96·10-s + 1.02·11-s − 0.175·12-s − 0.523·13-s + 0.699·14-s + 0.990·15-s − 1.21·16-s + 0.993·17-s + 0.380·18-s + 0.954·19-s − 0.522·20-s − 0.353·21-s + 1.17·22-s − 1.00·23-s + 0.458·24-s + 1.94·25-s − 0.597·26-s − 0.192·27-s + 0.186·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.943718275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.943718275\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 59 | \( 1 - 3.48e3T \) |
| good | 2 | \( 1 - 6.46T + 32T^{2} \) |
| 5 | \( 1 + 95.9T + 3.12e3T^{2} \) |
| 7 | \( 1 - 79.3T + 1.68e4T^{2} \) |
| 11 | \( 1 - 411.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 318.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.18e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.50e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.53e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.27e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.15e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.02e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.27e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 118.T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.71e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 5.53e3T + 4.18e8T^{2} \) |
| 61 | \( 1 - 4.48e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 8.93e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.17e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.03e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.55e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.17e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.06e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.66e4T + 8.58e9T^{2} \) |
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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
−11.91033255004480596775342533499, −11.53135101342254243790736129682, −10.00595342733563386398502622401, −8.495218134365777946862634775294, −7.51682666035040243472928608236, −6.32270117836956049771901056773, −4.94860574400873405364323027966, −4.25503826308913314868364532971, −3.23878009590516716951553053936, −0.78284424529553755152661852041,
0.78284424529553755152661852041, 3.23878009590516716951553053936, 4.25503826308913314868364532971, 4.94860574400873405364323027966, 6.32270117836956049771901056773, 7.51682666035040243472928608236, 8.495218134365777946862634775294, 10.00595342733563386398502622401, 11.53135101342254243790736129682, 11.91033255004480596775342533499
