L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.886 − 0.237i)3-s + (−0.866 + 0.499i)4-s + (−1.83 − 1.28i)5-s − 0.917i·6-s + (−2.44 − 1.00i)7-s + (−0.707 − 0.707i)8-s + (−1.86 − 1.07i)9-s + (0.767 − 2.10i)10-s + (1.86 + 2.74i)11-s + (0.886 − 0.237i)12-s + (2.12 + 2.12i)13-s + (0.334 − 2.62i)14-s + (1.31 + 1.57i)15-s + (0.500 − 0.866i)16-s + (5.54 + 1.48i)17-s + ⋯ |
L(s) = 1 | + (0.183 + 0.683i)2-s + (−0.511 − 0.137i)3-s + (−0.433 + 0.249i)4-s + (−0.818 − 0.574i)5-s − 0.374i·6-s + (−0.925 − 0.378i)7-s + (−0.249 − 0.249i)8-s + (−0.623 − 0.359i)9-s + (0.242 − 0.664i)10-s + (0.562 + 0.826i)11-s + (0.255 − 0.0685i)12-s + (0.588 + 0.588i)13-s + (0.0893 − 0.701i)14-s + (0.339 + 0.406i)15-s + (0.125 − 0.216i)16-s + (1.34 + 0.360i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.540 - 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.540 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.829129 + 0.452651i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.829129 + 0.452651i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 - 0.965i)T \) |
| 5 | \( 1 + (1.83 + 1.28i)T \) |
| 7 | \( 1 + (2.44 + 1.00i)T \) |
| 11 | \( 1 + (-1.86 - 2.74i)T \) |
good | 3 | \( 1 + (0.886 + 0.237i)T + (2.59 + 1.5i)T^{2} \) |
| 13 | \( 1 + (-2.12 - 2.12i)T + 13iT^{2} \) |
| 17 | \( 1 + (-5.54 - 1.48i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-1.63 + 2.83i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.48 - 5.55i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 - 1.08T + 29T^{2} \) |
| 31 | \( 1 + (-0.726 - 1.25i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.52 + 1.48i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 9.05iT - 41T^{2} \) |
| 43 | \( 1 + (4.48 + 4.48i)T + 43iT^{2} \) |
| 47 | \( 1 + (-10.7 + 2.86i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (4.26 + 1.14i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (5.17 - 2.98i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.666 + 0.384i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.31 - 12.3i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 8.83T + 71T^{2} \) |
| 73 | \( 1 + (1.38 - 5.15i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-4.54 + 7.87i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-9.94 - 9.94i)T + 83iT^{2} \) |
| 89 | \( 1 + (1.27 + 0.733i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.76 - 7.76i)T + 97iT^{2} \) |
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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
−10.40675368958732885588339743069, −9.297383155401248319138172214072, −8.849834197017788671211527089274, −7.57816824345755632967024762861, −7.02747650357170737813095287228, −6.06984235573361661419772669584, −5.24122845836296102939397534952, −4.06116465243607734261335424644, −3.35657906755385718878965411280, −0.940002270292633388851752524424,
0.69396049042205545825372991657, 2.96959393424118550162730287617, 3.30230483720240239762377686158, 4.61747929519101526177011428864, 5.89758902642689864691664570147, 6.28076916329132714434518542156, 7.77323243557477234417498622478, 8.492615310819324534245707333402, 9.554325324657206716821151966750, 10.45280845933962556013714851512