| L(s) = 1 | − 1.44i·3-s + (−0.386 − 2.20i)5-s + 3.25i·7-s + 0.918·9-s − 1.32·11-s − 1.25i·13-s + (−3.17 + 0.557i)15-s − 4.62i·17-s − 3.37·19-s + 4.70·21-s − i·23-s + (−4.70 + 1.70i)25-s − 5.65i·27-s − 4.29·29-s − 2.37·31-s + ⋯ |
| L(s) = 1 | − 0.832i·3-s + (−0.172 − 0.984i)5-s + 1.23i·7-s + 0.306·9-s − 0.400·11-s − 0.349i·13-s + (−0.820 + 0.143i)15-s − 1.12i·17-s − 0.773·19-s + 1.02·21-s − 0.208i·23-s + (−0.940 + 0.340i)25-s − 1.08i·27-s − 0.797·29-s − 0.426·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.172i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9175554585\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9175554585\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.386 + 2.20i)T \) |
| 23 | \( 1 + iT \) |
| good | 3 | \( 1 + 1.44iT - 3T^{2} \) |
| 7 | \( 1 - 3.25iT - 7T^{2} \) |
| 11 | \( 1 + 1.32T + 11T^{2} \) |
| 13 | \( 1 + 1.25iT - 13T^{2} \) |
| 17 | \( 1 + 4.62iT - 17T^{2} \) |
| 19 | \( 1 + 3.37T + 19T^{2} \) |
| 29 | \( 1 + 4.29T + 29T^{2} \) |
| 31 | \( 1 + 2.37T + 31T^{2} \) |
| 37 | \( 1 - 5.74iT - 37T^{2} \) |
| 41 | \( 1 + 5.13T + 41T^{2} \) |
| 43 | \( 1 + 10.4iT - 43T^{2} \) |
| 47 | \( 1 + 6.06iT - 47T^{2} \) |
| 53 | \( 1 + 11.1iT - 53T^{2} \) |
| 59 | \( 1 + 2.72T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 + 6.60iT - 67T^{2} \) |
| 71 | \( 1 + 0.265T + 71T^{2} \) |
| 73 | \( 1 - 5.26iT - 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 - 2.02iT - 83T^{2} \) |
| 89 | \( 1 + 16.1T + 89T^{2} \) |
| 97 | \( 1 - 8.62iT - 97T^{2} \) |
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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
−8.663787281304964533761725750631, −8.246838493605503418334231726673, −7.32981392625354834439903155690, −6.57192787442759784922870884920, −5.47826996774929115135709483506, −5.07947484000182181317150769357, −3.87934130146832827806282540100, −2.53086394892170904639654184855, −1.73237659728718696894312982260, −0.32569038573252979955676796428,
1.65587104465489233999168153843, 3.04348554055847416525499616071, 4.08030592467990855697702880910, 4.23363541948283360693692602332, 5.61232193197620413808247124653, 6.55129712200727615836896735505, 7.28845340654590535222305520404, 7.87929181713382153462485583143, 8.969544705345570131597227408495, 9.908433083927681287748660331898
