L(s) = 1 | + (−0.207 − 0.358i)3-s − 2.82·5-s + (−0.792 + 1.37i)7-s + (1.41 − 2.44i)9-s + (2.62 + 4.54i)11-s + (−1 + 3.46i)13-s + (0.585 + 1.01i)15-s + (0.0857 − 0.148i)17-s + (−3.62 + 6.27i)19-s + 0.656·21-s + (3.62 + 6.27i)23-s + 3.00·25-s − 2.41·27-s + (1.32 + 2.30i)29-s − 5.65·31-s + ⋯ |
L(s) = 1 | + (−0.119 − 0.207i)3-s − 1.26·5-s + (−0.299 + 0.519i)7-s + (0.471 − 0.816i)9-s + (0.790 + 1.36i)11-s + (−0.277 + 0.960i)13-s + (0.151 + 0.261i)15-s + (0.0208 − 0.0360i)17-s + (−0.830 + 1.43i)19-s + 0.143·21-s + (0.755 + 1.30i)23-s + 0.600·25-s − 0.464·27-s + (0.246 + 0.427i)29-s − 1.01·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.576056 + 0.568716i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.576056 + 0.568716i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 + (1 - 3.46i)T \) |
good | 3 | \( 1 + (0.207 + 0.358i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + 2.82T + 5T^{2} \) |
| 7 | \( 1 + (0.792 - 1.37i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.62 - 4.54i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.0857 + 0.148i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.62 - 6.27i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.62 - 6.27i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.32 - 2.30i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 + (4.74 + 8.21i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.0857 - 0.148i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.03 + 8.72i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 2.82T + 53T^{2} \) |
| 59 | \( 1 + (3.62 - 6.27i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.37 + 4.11i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.621 - 1.07i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 4.48T + 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + (-7.32 - 12.6i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.5 + 7.79i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.80393691613890849795877881789, −10.60069393281928365062442909483, −9.408868780057483205130325782576, −8.902690870134243469155965645063, −7.36033793133985790473398221516, −7.11120684372679080316926938719, −5.81696401799114879105412283180, −4.26256629918312557751062620821, −3.71599974571601652190490860191, −1.76323639653083310184307739574,
0.54647633518433370662244596702, 2.96603595367112199857029046969, 4.04840041830719252535537556835, 4.92692760236152258859330306472, 6.38828120616388474176376717509, 7.33462485768820063904887853516, 8.214888612049930276306394267408, 8.999573009015406530439808706843, 10.42982403381762458570429438831, 10.92929978484628411864442055441