| L(s) = 1 | + (0.733 − 0.680i)2-s + (0.371 − 0.0559i)3-s + (0.0747 − 0.997i)4-s + (0.782 − 1.99i)5-s + (0.234 − 0.293i)6-s + (−0.623 − 0.781i)8-s + (−2.73 + 0.842i)9-s + (−0.782 − 1.99i)10-s + (−2.04 − 0.631i)11-s + (−0.0280 − 0.374i)12-s + (−1.27 − 5.57i)13-s + (0.178 − 0.783i)15-s + (−0.988 − 0.149i)16-s + (4.05 − 2.76i)17-s + (−1.42 + 2.47i)18-s + (1.03 + 1.79i)19-s + ⋯ |
| L(s) = 1 | + (0.518 − 0.480i)2-s + (0.214 − 0.0323i)3-s + (0.0373 − 0.498i)4-s + (0.349 − 0.891i)5-s + (0.0955 − 0.119i)6-s + (−0.220 − 0.276i)8-s + (−0.910 + 0.280i)9-s + (−0.247 − 0.630i)10-s + (−0.617 − 0.190i)11-s + (−0.00810 − 0.108i)12-s + (−0.352 − 1.54i)13-s + (0.0462 − 0.202i)15-s + (−0.247 − 0.0372i)16-s + (0.982 − 0.669i)17-s + (−0.336 + 0.583i)18-s + (0.237 + 0.411i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 686 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 + 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 686 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.845777 - 1.65085i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.845777 - 1.65085i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.733 + 0.680i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (-0.371 + 0.0559i)T + (2.86 - 0.884i)T^{2} \) |
| 5 | \( 1 + (-0.782 + 1.99i)T + (-3.66 - 3.40i)T^{2} \) |
| 11 | \( 1 + (2.04 + 0.631i)T + (9.08 + 6.19i)T^{2} \) |
| 13 | \( 1 + (1.27 + 5.57i)T + (-11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (-4.05 + 2.76i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (-1.03 - 1.79i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (5.23 + 3.56i)T + (8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (-7.15 - 3.44i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (-3.30 + 5.72i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.306 - 4.09i)T + (-36.5 + 5.51i)T^{2} \) |
| 41 | \( 1 + (-3.90 - 4.89i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (3.22 - 4.04i)T + (-9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (0.900 - 0.835i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (-0.571 + 7.62i)T + (-52.4 - 7.89i)T^{2} \) |
| 59 | \( 1 + (1.00 + 2.56i)T + (-43.2 + 40.1i)T^{2} \) |
| 61 | \( 1 + (0.705 + 9.41i)T + (-60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (0.196 - 0.341i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-11.3 + 5.46i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-3.16 - 2.93i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (-6.07 - 10.5i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.51 - 6.63i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (0.929 - 0.286i)T + (73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 - 3.19T + 97T^{2} \) |
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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
−10.13027516873651362149366722737, −9.573599165313665775173512690315, −8.167594358941968618182554977809, −8.040071384265853556626499239422, −6.26878419557013145477835967265, −5.34805521110624897049529591800, −4.90605210825771899204516050732, −3.30280876786927716522096034740, −2.50549737006365514522117128099, −0.798934941946932352166530844299,
2.24628417216668174223992198025, 3.19085304434073234342497568417, 4.31890387121708274223156205624, 5.54268221733816449648286811480, 6.31515933626000625677836018788, 7.12259839354572567131281238333, 8.044669664684779280345761754305, 8.968628405493553892715599572370, 9.944021454785428426863623168573, 10.73784220915661179410912259278
