| L(s) = 1 | + (−0.987 − 0.156i)2-s + (0.992 − 1.94i)3-s + (0.951 + 0.309i)4-s + (0.244 − 2.22i)5-s + (−1.28 + 1.76i)6-s + (−3.05 + 1.55i)7-s + (−0.891 − 0.453i)8-s + (−1.04 − 1.43i)9-s + (−0.589 + 2.15i)10-s + (3.24 − 0.694i)11-s + (1.54 − 1.54i)12-s + (0.387 − 2.44i)13-s + (3.26 − 1.06i)14-s + (−4.08 − 2.68i)15-s + (0.809 + 0.587i)16-s + (−0.196 − 1.24i)17-s + ⋯ |
| L(s) = 1 | + (−0.698 − 0.110i)2-s + (0.572 − 1.12i)3-s + (0.475 + 0.154i)4-s + (0.109 − 0.994i)5-s + (−0.524 + 0.721i)6-s + (−1.15 + 0.588i)7-s + (−0.315 − 0.160i)8-s + (−0.347 − 0.478i)9-s + (−0.186 + 0.682i)10-s + (0.977 − 0.209i)11-s + (0.446 − 0.446i)12-s + (0.107 − 0.679i)13-s + (0.872 − 0.283i)14-s + (−1.05 − 0.692i)15-s + (0.202 + 0.146i)16-s + (−0.0477 − 0.301i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0886 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0886 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.642297 - 0.587682i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.642297 - 0.587682i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.987 + 0.156i)T \) |
| 5 | \( 1 + (-0.244 + 2.22i)T \) |
| 11 | \( 1 + (-3.24 + 0.694i)T \) |
| good | 3 | \( 1 + (-0.992 + 1.94i)T + (-1.76 - 2.42i)T^{2} \) |
| 7 | \( 1 + (3.05 - 1.55i)T + (4.11 - 5.66i)T^{2} \) |
| 13 | \( 1 + (-0.387 + 2.44i)T + (-12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (0.196 + 1.24i)T + (-16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (-1.62 - 5.00i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-5.42 - 5.42i)T + 23iT^{2} \) |
| 29 | \( 1 + (-0.472 + 1.45i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (6.34 - 4.61i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.60 - 5.11i)T + (-21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (-3.98 + 1.29i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-0.832 + 0.832i)T - 43iT^{2} \) |
| 47 | \( 1 + (4.97 + 2.53i)T + (27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (9.00 + 1.42i)T + (50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (-2.65 - 0.862i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-6.34 + 8.73i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (4.03 - 4.03i)T - 67iT^{2} \) |
| 71 | \( 1 + (7.25 + 5.27i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.14 - 8.12i)T + (-42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (11.2 - 8.19i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (11.9 - 1.89i)T + (78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + 4.89iT - 89T^{2} \) |
| 97 | \( 1 + (-0.442 + 2.79i)T + (-92.2 - 29.9i)T^{2} \) |
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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
−13.03649737905573623508886899768, −12.61906257844318263271024603321, −11.60094244458260487703236581464, −9.800438618276861323951034413061, −9.029876465736283916127371204737, −8.115129283331720781366293968765, −6.94994301522794341668895670237, −5.73488695716220337464837260059, −3.22947545964520374859585120255, −1.40732525469904195243511262032,
2.96701524217947346251993523377, 4.14140845708374506621238262687, 6.42847480788112744267540732725, 7.17933378924105675955591061824, 9.066854972325833087731270017867, 9.509272920363315537842372836588, 10.49839990327176361086795421828, 11.31607777373193951665456246438, 13.00589256882360767528595845365, 14.37619121399570561554099359222
