| L(s) = 1 | + (−2.39 + 1.06i)2-s + (0.360 + 1.69i)3-s + (3.24 − 3.60i)4-s + (2.95 + 1.31i)5-s + (−2.66 − 3.66i)6-s + (0.978 + 0.207i)7-s + (−2.30 + 7.10i)8-s + (−2.74 + 1.22i)9-s − 8.47·10-s + (−2.38 − 2.30i)11-s + (7.28 + 4.20i)12-s + (−0.258 + 2.45i)13-s + (−2.56 + 0.544i)14-s + (−1.16 + 5.48i)15-s + (−1.03 − 9.80i)16-s + (1.5 − 1.08i)17-s + ⋯ |
| L(s) = 1 | + (−1.69 + 0.752i)2-s + (0.207 + 0.978i)3-s + (1.62 − 1.80i)4-s + (1.32 + 0.588i)5-s + (−1.08 − 1.49i)6-s + (0.369 + 0.0785i)7-s + (−0.816 + 2.51i)8-s + (−0.913 + 0.406i)9-s − 2.67·10-s + (−0.717 − 0.696i)11-s + (2.10 + 1.21i)12-s + (−0.0716 + 0.681i)13-s + (−0.684 + 0.145i)14-s + (−0.300 + 1.41i)15-s + (−0.257 − 2.45i)16-s + (0.363 − 0.264i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.351709 + 0.495302i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.351709 + 0.495302i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.360 - 1.69i)T \) |
| 11 | \( 1 + (2.38 + 2.30i)T \) |
| good | 2 | \( 1 + (2.39 - 1.06i)T + (1.33 - 1.48i)T^{2} \) |
| 5 | \( 1 + (-2.95 - 1.31i)T + (3.34 + 3.71i)T^{2} \) |
| 7 | \( 1 + (-0.978 - 0.207i)T + (6.39 + 2.84i)T^{2} \) |
| 13 | \( 1 + (0.258 - 2.45i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (-1.5 + 1.08i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.309 + 0.951i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (1.19 + 2.06i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.81 + 0.385i)T + (26.4 + 11.7i)T^{2} \) |
| 31 | \( 1 + (-0.169 + 1.60i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-1.88 - 5.79i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-11.2 + 2.38i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-2.42 + 4.20i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3.97 + 4.41i)T + (-4.91 + 46.7i)T^{2} \) |
| 53 | \( 1 + (-5.73 - 4.16i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (0.413 - 0.459i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (0.433 + 4.12i)T + (-59.6 + 12.6i)T^{2} \) |
| 67 | \( 1 + (3 + 5.19i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-11.3 + 8.28i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (2.11 + 6.51i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (8.73 - 3.88i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-1.48 - 14.1i)T + (-81.1 + 17.2i)T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + (-5.48 + 2.44i)T + (64.9 - 72.0i)T^{2} \) |
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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
−14.52792480560273714358527364016, −13.78962407554264520287355011027, −11.29668953224283084280513483841, −10.52060011433752043937799261572, −9.743674920337018349641530924504, −8.976034656473504248295718813515, −7.86266675230813577644449648855, −6.36981426161494575469330794501, −5.39467360116086427635480141781, −2.39534913888368332658669380092,
1.40426069247133108431275174212, 2.55966398499463621512819833532, 5.78262692902104318883977465513, 7.38836171951280720471647718363, 8.194441375470245257483212747697, 9.307893521162440938250664734089, 10.09655023060878248103820480215, 11.24819015708460243145797846702, 12.57996847248017842786876847558, 13.00580953597126243180880204158
