| L(s) = 1 | + (−0.891 − 0.453i)2-s + (2.03 − 0.322i)3-s + (0.587 + 0.809i)4-s + (0.667 + 2.13i)5-s + (−1.95 − 0.636i)6-s + (−0.0611 + 0.386i)7-s + (−0.156 − 0.987i)8-s + (1.18 − 0.384i)9-s + (0.373 − 2.20i)10-s + (−3.24 − 0.696i)11-s + (1.45 + 1.45i)12-s + (2.60 − 5.11i)13-s + (0.229 − 0.316i)14-s + (2.04 + 4.12i)15-s + (−0.309 + 0.951i)16-s + (0.224 + 0.440i)17-s + ⋯ |
| L(s) = 1 | + (−0.630 − 0.321i)2-s + (1.17 − 0.186i)3-s + (0.293 + 0.404i)4-s + (0.298 + 0.954i)5-s + (−0.799 − 0.259i)6-s + (−0.0231 + 0.145i)7-s + (−0.0553 − 0.349i)8-s + (0.394 − 0.128i)9-s + (0.118 − 0.697i)10-s + (−0.977 − 0.210i)11-s + (0.420 + 0.420i)12-s + (0.723 − 1.41i)13-s + (0.0614 − 0.0845i)14-s + (0.528 + 1.06i)15-s + (−0.0772 + 0.237i)16-s + (0.0544 + 0.106i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.117i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.07938 - 0.0638108i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.07938 - 0.0638108i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.891 + 0.453i)T \) |
| 5 | \( 1 + (-0.667 - 2.13i)T \) |
| 11 | \( 1 + (3.24 + 0.696i)T \) |
| good | 3 | \( 1 + (-2.03 + 0.322i)T + (2.85 - 0.927i)T^{2} \) |
| 7 | \( 1 + (0.0611 - 0.386i)T + (-6.65 - 2.16i)T^{2} \) |
| 13 | \( 1 + (-2.60 + 5.11i)T + (-7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (-0.224 - 0.440i)T + (-9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (2.61 + 1.90i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.05 + 1.05i)T - 23iT^{2} \) |
| 29 | \( 1 + (6.85 - 4.97i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (1.58 + 4.87i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-9.20 - 1.45i)T + (35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (3.50 - 4.82i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-3.86 - 3.86i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.596 + 3.76i)T + (-44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (-1.62 - 0.830i)T + (31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (-1.77 - 2.44i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-7.86 - 2.55i)T + (49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (10.6 + 10.6i)T + 67iT^{2} \) |
| 71 | \( 1 + (-3.83 + 11.8i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (11.9 + 1.89i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-2.96 - 9.12i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-5.29 + 2.69i)T + (48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 - 9.83iT - 89T^{2} \) |
| 97 | \( 1 + (2.44 - 4.79i)T + (-57.0 - 78.4i)T^{2} \) |
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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
−13.43952276906345550151654621783, −12.97716872138964401332882186835, −11.15538681681899291569113915461, −10.47987068475535631605964019492, −9.303538530284405104591345527125, −8.205262721562147466666551457282, −7.48114539418584403252172092549, −5.88026588475879062335899655039, −3.34113885284566105267708371976, −2.44260292204020476882042747305,
2.08148055757081211289421928843, 4.12617195148352063365904496021, 5.78051010331515263249155063046, 7.43960283378527250784254966547, 8.523228528158739901560571248815, 9.115742336043115423668713200075, 10.07639961443334041229606356466, 11.48610002713363460454386284923, 12.97135781593774928720994717738, 13.77437361568574045183165830826
