L(s) = 1 | + (1.38 − 0.303i)2-s + (2.07 − 0.554i)3-s + (1.81 − 0.837i)4-s + (0.992 + 0.265i)5-s + (2.69 − 1.39i)6-s + (2.25 − 1.70i)8-s + (1.38 − 0.797i)9-s + (1.45 + 0.0664i)10-s + (−0.340 − 1.27i)11-s + (3.29 − 2.74i)12-s + (−3.98 − 3.98i)13-s + 2.20·15-s + (2.59 − 3.04i)16-s + (−2.81 + 4.87i)17-s + (1.66 − 1.52i)18-s + (−1.25 + 4.69i)19-s + ⋯ |
L(s) = 1 | + (0.976 − 0.214i)2-s + (1.19 − 0.320i)3-s + (0.908 − 0.418i)4-s + (0.443 + 0.118i)5-s + (1.09 − 0.569i)6-s + (0.797 − 0.603i)8-s + (0.460 − 0.265i)9-s + (0.458 + 0.0210i)10-s + (−0.102 − 0.383i)11-s + (0.951 − 0.791i)12-s + (−1.10 − 1.10i)13-s + 0.568·15-s + (0.649 − 0.760i)16-s + (−0.682 + 1.18i)17-s + (0.392 − 0.358i)18-s + (−0.288 + 1.07i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 + 0.652i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.757 + 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.86471 - 1.43476i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.86471 - 1.43476i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.38 + 0.303i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-2.07 + 0.554i)T + (2.59 - 1.5i)T^{2} \) |
| 5 | \( 1 + (-0.992 - 0.265i)T + (4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (0.340 + 1.27i)T + (-9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (3.98 + 3.98i)T + 13iT^{2} \) |
| 17 | \( 1 + (2.81 - 4.87i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.25 - 4.69i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-0.495 + 0.286i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.81 - 5.81i)T + 29iT^{2} \) |
| 31 | \( 1 + (1.27 - 2.20i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-7.52 - 2.01i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 4.12iT - 41T^{2} \) |
| 43 | \( 1 + (-0.783 + 0.783i)T - 43iT^{2} \) |
| 47 | \( 1 + (-4.77 - 8.26i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.26 + 4.72i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (3.71 + 13.8i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-2.32 + 8.68i)T + (-52.8 - 30.5i)T^{2} \) |
| 67 | \( 1 + (8.24 - 2.20i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 9.57iT - 71T^{2} \) |
| 73 | \( 1 + (10.9 + 6.32i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.29 - 7.43i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.65 - 2.65i)T + 83iT^{2} \) |
| 89 | \( 1 + (-2.44 + 1.41i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 8.98T + 97T^{2} \) |
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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
−10.31045516025883586740350227760, −9.455189238990190294365736080874, −8.206045797230451050471755786108, −7.77426270473196036024194409927, −6.55308275166769349893536232878, −5.77237378188493841063640116250, −4.65866765417401896516626274917, −3.45149074340568411643529389875, −2.68778851847386506286903251538, −1.73903755255325933565755507800,
2.31011143749517332209666192964, 2.63950200377390653579518149418, 4.17306875347413676478798947335, 4.67007659466170164492819051567, 5.89389034435361632501407434354, 7.05081516899288673859596345969, 7.56203415615003686642894491120, 8.833917183116002010835028862889, 9.377300024744858355415937333743, 10.25186484639495934126801055567