| L(s) = 1 | + (0.997 − 0.0713i)2-s + (−0.307 + 1.41i)3-s + (0.989 − 0.142i)4-s + (1.55 − 1.60i)5-s + (−0.205 + 1.43i)6-s + (−1.84 + 3.38i)7-s + (0.977 − 0.212i)8-s + (0.829 + 0.378i)9-s + (1.43 − 1.71i)10-s + (3.85 − 3.34i)11-s + (−0.103 + 1.44i)12-s + (−3.94 + 2.15i)13-s + (−1.59 + 3.50i)14-s + (1.79 + 2.68i)15-s + (0.959 − 0.281i)16-s + (−2.64 − 3.52i)17-s + ⋯ |
| L(s) = 1 | + (0.705 − 0.0504i)2-s + (−0.177 + 0.815i)3-s + (0.494 − 0.0711i)4-s + (0.695 − 0.718i)5-s + (−0.0839 + 0.583i)6-s + (−0.697 + 1.27i)7-s + (0.345 − 0.0751i)8-s + (0.276 + 0.126i)9-s + (0.454 − 0.541i)10-s + (1.16 − 1.00i)11-s + (−0.0297 + 0.416i)12-s + (−1.09 + 0.597i)13-s + (−0.427 + 0.936i)14-s + (0.462 + 0.694i)15-s + (0.239 − 0.0704i)16-s + (−0.640 − 0.855i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 - 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.826 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.76206 + 0.543602i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.76206 + 0.543602i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.997 + 0.0713i)T \) |
| 5 | \( 1 + (-1.55 + 1.60i)T \) |
| 23 | \( 1 + (-2.37 + 4.16i)T \) |
| good | 3 | \( 1 + (0.307 - 1.41i)T + (-2.72 - 1.24i)T^{2} \) |
| 7 | \( 1 + (1.84 - 3.38i)T + (-3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (-3.85 + 3.34i)T + (1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (3.94 - 2.15i)T + (7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (2.64 + 3.52i)T + (-4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (-0.596 - 4.15i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (5.29 + 0.761i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-2.10 - 1.35i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (3.94 + 10.5i)T + (-27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (2.05 + 4.50i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (3.43 + 0.746i)T + (39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (4.84 + 4.84i)T + 47iT^{2} \) |
| 53 | \( 1 + (-8.03 - 4.38i)T + (28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (-0.493 + 1.68i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (0.855 - 1.33i)T + (-25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-0.499 - 6.98i)T + (-66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (9.09 - 10.4i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-3.61 - 2.70i)T + (20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (-4.81 - 1.41i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-5.41 + 2.01i)T + (62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (-0.368 + 0.237i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-16.6 - 6.19i)T + (73.3 + 63.5i)T^{2} \) |
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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
−12.26339006382349269653710317522, −11.62423629388928847245753685173, −10.29961024490091704922073395826, −9.318916389567528410844354722746, −8.870314418507662024905151428025, −6.88415543726152518170922785740, −5.80056329584370508722648710877, −5.04166256818727271483394464369, −3.84276555853810544889006982853, −2.24457358513597883647837372696,
1.71094987056408626343925137612, 3.36815044528985104440841612304, 4.67701864825192254488267288859, 6.34935152660997419491653203276, 6.90382607526158386378440573827, 7.46085728083772044045162092125, 9.576328559029119994660087987254, 10.16731879799761658516832123338, 11.33350719462527212954276648561, 12.36383565291186061429326714128
