| L(s) = 1 | + (0.186 − 0.215i)2-s + (−2.03 + 1.31i)3-s + (0.273 + 1.89i)4-s + (−0.415 − 0.909i)5-s + (−0.0982 + 0.683i)6-s + (−4.87 + 1.43i)7-s + (0.938 + 0.603i)8-s + (1.19 − 2.61i)9-s + (−0.273 − 0.0801i)10-s + (2.39 + 2.76i)11-s + (−3.04 − 3.51i)12-s + (4.71 + 1.38i)13-s + (−0.600 + 1.31i)14-s + (2.03 + 1.31i)15-s + (−3.37 + 0.991i)16-s + (0.237 − 1.65i)17-s + ⋯ |
| L(s) = 1 | + (0.131 − 0.152i)2-s + (−1.17 + 0.756i)3-s + (0.136 + 0.949i)4-s + (−0.185 − 0.406i)5-s + (−0.0400 + 0.278i)6-s + (−1.84 + 0.540i)7-s + (0.331 + 0.213i)8-s + (0.398 − 0.872i)9-s + (−0.0863 − 0.0253i)10-s + (0.722 + 0.834i)11-s + (−0.879 − 1.01i)12-s + (1.30 + 0.384i)13-s + (−0.160 + 0.351i)14-s + (0.526 + 0.338i)15-s + (−0.844 + 0.247i)16-s + (0.0576 − 0.400i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.464 - 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.464 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.320958 + 0.530751i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.320958 + 0.530751i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.415 + 0.909i)T \) |
| 23 | \( 1 + (-3.82 - 2.89i)T \) |
| good | 2 | \( 1 + (-0.186 + 0.215i)T + (-0.284 - 1.97i)T^{2} \) |
| 3 | \( 1 + (2.03 - 1.31i)T + (1.24 - 2.72i)T^{2} \) |
| 7 | \( 1 + (4.87 - 1.43i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (-2.39 - 2.76i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-4.71 - 1.38i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.237 + 1.65i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (0.314 + 2.18i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (0.240 - 1.67i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (1.04 + 0.669i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (1.32 - 2.91i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (-3.16 - 6.92i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (1.68 - 1.08i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 - 8.67T + 47T^{2} \) |
| 53 | \( 1 + (5.20 - 1.52i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (1.56 + 0.460i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (6.36 + 4.09i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-0.0752 + 0.0868i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (1.26 - 1.46i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-1.48 - 10.3i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (0.948 + 0.278i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-6.48 + 14.1i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (-2.51 + 1.61i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (4.59 + 10.0i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−13.45490628700584857891022759112, −12.64731650942296103391391271191, −11.87402266511518611996564527020, −11.04936404801224231505973923664, −9.650074023814972111901580420156, −8.909278240626428716579755833153, −6.98936899047419853878066215717, −6.04353444264941170180221695508, −4.50163073278008411277925610140, −3.34413516324959046072083850938,
0.77841955954852869248940946531, 3.64366035660275821600007855062, 5.89930652547945391329457697301, 6.25725862044391546762099844097, 7.09714937196649839484743692958, 9.117111565429005120012859888371, 10.52189338507206167291793581572, 10.94830580303303754080131985138, 12.27076138897836245368521419310, 13.21494837713401396594230887022
