L(s) = 1 | + (0.959 − 0.281i)2-s + (0.251 + 1.74i)3-s + (0.841 − 0.540i)4-s + (0.415 − 0.909i)5-s + (0.732 + 1.60i)6-s + (−0.408 + 0.120i)7-s + (0.654 − 0.755i)8-s + (−0.108 + 0.0317i)9-s + (0.142 − 0.989i)10-s + (−1.34 + 2.95i)11-s + (1.15 + 1.33i)12-s + (0.852 + 0.983i)13-s + (−0.358 + 0.230i)14-s + (1.69 + 0.497i)15-s + (0.415 − 0.909i)16-s + (5.01 + 3.22i)17-s + ⋯ |
L(s) = 1 | + (0.678 − 0.199i)2-s + (0.144 + 1.00i)3-s + (0.420 − 0.270i)4-s + (0.185 − 0.406i)5-s + (0.299 + 0.655i)6-s + (−0.154 + 0.0453i)7-s + (0.231 − 0.267i)8-s + (−0.0360 + 0.0105i)9-s + (0.0450 − 0.313i)10-s + (−0.406 + 0.890i)11-s + (0.333 + 0.384i)12-s + (0.236 + 0.272i)13-s + (−0.0958 + 0.0615i)14-s + (0.437 + 0.128i)15-s + (0.103 − 0.227i)16-s + (1.21 + 0.781i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.36803 + 0.801071i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.36803 + 0.801071i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.959 + 0.281i)T \) |
| 5 | \( 1 + (-0.415 + 0.909i)T \) |
| 67 | \( 1 + (-7.43 + 3.42i)T \) |
good | 3 | \( 1 + (-0.251 - 1.74i)T + (-2.87 + 0.845i)T^{2} \) |
| 7 | \( 1 + (0.408 - 0.120i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (1.34 - 2.95i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.852 - 0.983i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-5.01 - 3.22i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-6.42 - 1.88i)T + (15.9 + 10.2i)T^{2} \) |
| 23 | \( 1 + (0.740 + 5.15i)T + (-22.0 + 6.47i)T^{2} \) |
| 29 | \( 1 + 6.66T + 29T^{2} \) |
| 31 | \( 1 + (-1.91 + 2.20i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + 7.38T + 37T^{2} \) |
| 41 | \( 1 + (5.28 + 3.39i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (-2.52 - 1.62i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + (-0.330 - 2.29i)T + (-45.0 + 13.2i)T^{2} \) |
| 53 | \( 1 + (-9.48 + 6.09i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-3.93 + 4.54i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (4.13 + 9.05i)T + (-39.9 + 46.1i)T^{2} \) |
| 71 | \( 1 + (12.0 - 7.75i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (2.96 + 6.48i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (3.75 + 4.33i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (4.13 - 9.04i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (-2.26 + 15.7i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 - 0.475T + 97T^{2} \) |
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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
−10.31447624599767279296043616591, −9.995001608833374744728424037528, −9.171233693417639556147661502050, −7.997068541040951017908543950753, −6.98973289852551177888987129821, −5.71627224160454626371791675456, −5.02169825291225263324929321017, −4.06675246914695875789439949210, −3.26110557704820184873094568457, −1.68763416843878127056452667098,
1.27903348983039635953863476326, 2.80562458396070526545410702197, 3.55607251072676264983665491971, 5.28937857947242504974060151788, 5.81386265255458194087143150032, 7.11104714414467316764708106137, 7.39730563929396921517084457588, 8.370806085523956911982928478052, 9.621749140766967444936719745584, 10.51625123919094536763085395832