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
−13.21494837713401396594230887022, −12.27076138897836245368521419310, −10.94830580303303754080131985138, −10.52189338507206167291793581572, −9.117111565429005120012859888371, −7.09714937196649839484743692958, −6.25725862044391546762099844097, −5.89930652547945391329457697301, −3.64366035660275821600007855062, −0.77841955954852869248940946531,
3.34413516324959046072083850938, 4.50163073278008411277925610140, 6.04353444264941170180221695508, 6.98936899047419853878066215717, 8.909278240626428716579755833153, 9.650074023814972111901580420156, 11.04936404801224231505973923664, 11.87402266511518611996564527020, 12.64731650942296103391391271191, 13.45490628700584857891022759112