| L(s) = 1 | + (0.939 − 0.342i)2-s + (0.552 − 3.13i)3-s + (0.766 − 0.642i)4-s + (0.766 + 0.642i)5-s + (−0.552 − 3.13i)6-s + (1.67 + 2.89i)7-s + (0.500 − 0.866i)8-s + (−6.68 − 2.43i)9-s + (0.939 + 0.342i)10-s + (−3.09 + 5.35i)11-s + (−1.58 − 2.75i)12-s + (−0.128 − 0.727i)13-s + (2.56 + 2.15i)14-s + (2.43 − 2.04i)15-s + (0.173 − 0.984i)16-s + (−0.815 + 0.296i)17-s + ⋯ |
| L(s) = 1 | + (0.664 − 0.241i)2-s + (0.318 − 1.80i)3-s + (0.383 − 0.321i)4-s + (0.342 + 0.287i)5-s + (−0.225 − 1.27i)6-s + (0.632 + 1.09i)7-s + (0.176 − 0.306i)8-s + (−2.22 − 0.810i)9-s + (0.297 + 0.108i)10-s + (−0.932 + 1.61i)11-s + (−0.458 − 0.794i)12-s + (−0.0355 − 0.201i)13-s + (0.685 + 0.574i)14-s + (0.628 − 0.527i)15-s + (0.0434 − 0.246i)16-s + (−0.197 + 0.0720i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.153 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.153 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.41253 - 1.20989i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.41253 - 1.20989i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.939 + 0.342i)T \) |
| 5 | \( 1 + (-0.766 - 0.642i)T \) |
| 19 | \( 1 + (2.59 + 3.49i)T \) |
| good | 3 | \( 1 + (-0.552 + 3.13i)T + (-2.81 - 1.02i)T^{2} \) |
| 7 | \( 1 + (-1.67 - 2.89i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (3.09 - 5.35i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.128 + 0.727i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (0.815 - 0.296i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (0.875 - 0.735i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-7.32 - 2.66i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (3.23 + 5.60i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 1.68T + 37T^{2} \) |
| 41 | \( 1 + (-1.68 + 9.54i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-1.42 - 1.19i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (0.311 + 0.113i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (5.44 - 4.56i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (4.56 - 1.66i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-5.53 + 4.64i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (2.74 + 1.00i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (5.27 + 4.42i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-1.35 + 7.69i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (0.399 - 2.26i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-5.72 - 9.91i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.404 - 2.29i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-6.43 + 2.34i)T + (74.3 - 62.3i)T^{2} \) |
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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
−12.51729227603227476382467688435, −11.85470053471713362043538908840, −10.70399584746507087867410048972, −9.161942313549083141985882675111, −7.973551088283091595785606838706, −7.12274249735815805489234003884, −6.09857470540345549732749629309, −4.98757254420053467030023846200, −2.56746325981405006283886389191, −2.01332324628375114090870676913,
3.04637925271467669850723554427, 4.17909346323554715262651451633, 4.99424262789684390254347485610, 6.06205267520292137297674257416, 8.004035979548521735804470869964, 8.682711126322784627657448961145, 10.11573266787273703488368899881, 10.70332950116476708161736982484, 11.46450012838157336785502908999, 13.14094628232349220847130900726
