L(s) = 1 | + (−0.953 + 1.04i)2-s + (0.554 + 2.07i)3-s + (−0.182 − 1.99i)4-s + (0.265 − 0.992i)5-s + (−2.69 − 1.39i)6-s + (2.25 + 1.70i)8-s + (−1.38 + 0.797i)9-s + (0.783 + 1.22i)10-s + (1.27 − 0.340i)11-s + (4.02 − 1.48i)12-s + (3.98 − 3.98i)13-s + 2.20·15-s + (−3.93 + 0.728i)16-s + (2.81 − 4.87i)17-s + (0.483 − 2.20i)18-s + (−4.69 − 1.25i)19-s + ⋯ |
L(s) = 1 | + (−0.673 + 0.738i)2-s + (0.320 + 1.19i)3-s + (−0.0914 − 0.995i)4-s + (0.118 − 0.443i)5-s + (−1.09 − 0.569i)6-s + (0.797 + 0.603i)8-s + (−0.460 + 0.265i)9-s + (0.247 + 0.386i)10-s + (0.383 − 0.102i)11-s + (1.16 − 0.428i)12-s + (1.10 − 1.10i)13-s + 0.568·15-s + (−0.983 + 0.182i)16-s + (0.682 − 1.18i)17-s + (0.114 − 0.519i)18-s + (−1.07 − 0.288i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 - 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.18422 + 0.619874i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.18422 + 0.619874i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.953 - 1.04i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.554 - 2.07i)T + (-2.59 + 1.5i)T^{2} \) |
| 5 | \( 1 + (-0.265 + 0.992i)T + (-4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (-1.27 + 0.340i)T + (9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (-3.98 + 3.98i)T - 13iT^{2} \) |
| 17 | \( 1 + (-2.81 + 4.87i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.69 + 1.25i)T + (16.4 + 9.5i)T^{2} \) |
| 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 | \( 1 + (2.01 - 7.52i)T + (-32.0 - 18.5i)T^{2} \) |
| 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 | \( 1 + (-4.72 + 1.26i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (13.8 - 3.71i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-8.68 - 2.32i)T + (52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (-2.20 - 8.24i)T + (-58.0 + 33.5i)T^{2} \) |
| 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
−10.09190527186036673644819503305, −9.608272356461433302189677795971, −8.585265874263310106480985519097, −8.328380281883937722627315364755, −6.99566209812057184630824854479, −5.99426032818278274208896534180, −5.07919756243945816077458185433, −4.28126532369919340680786422085, −2.98608070704967473650662131170, −0.999554444962250372146755556625,
1.31410936928735187446382661222, 2.05492888712370984558840584924, 3.32634111017398687071039171285, 4.34578372613042082304172207146, 6.32634366855646696367310015996, 6.72959245620925039530083662793, 7.76465343639459708750245408950, 8.529948478508280658582121633739, 9.089210281886850795931968215836, 10.45575075270461104183564727433