| L(s) = 1 | + (0.991 − 0.130i)2-s + (−0.442 − 0.896i)3-s + (0.965 − 0.258i)4-s + (2.15 − 1.88i)5-s + (−0.555 − 0.831i)6-s + (0.334 − 2.62i)7-s + (0.923 − 0.382i)8-s + (−0.608 + 0.793i)9-s + (1.88 − 2.15i)10-s + (−1.18 + 0.0777i)11-s + (−0.659 − 0.751i)12-s + (−2.39 − 2.39i)13-s + (−0.0108 − 2.64i)14-s + (−2.64 − 1.09i)15-s + (0.866 − 0.5i)16-s + (−3.36 − 2.38i)17-s + ⋯ |
| L(s) = 1 | + (0.701 − 0.0922i)2-s + (−0.255 − 0.517i)3-s + (0.482 − 0.129i)4-s + (0.963 − 0.844i)5-s + (−0.226 − 0.339i)6-s + (0.126 − 0.991i)7-s + (0.326 − 0.135i)8-s + (−0.202 + 0.264i)9-s + (0.597 − 0.681i)10-s + (−0.357 + 0.0234i)11-s + (−0.190 − 0.217i)12-s + (−0.664 − 0.664i)13-s + (−0.00290 − 0.707i)14-s + (−0.683 − 0.283i)15-s + (0.216 − 0.125i)16-s + (−0.816 − 0.577i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.54242 - 1.76827i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.54242 - 1.76827i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.991 + 0.130i)T \) |
| 3 | \( 1 + (0.442 + 0.896i)T \) |
| 7 | \( 1 + (-0.334 + 2.62i)T \) |
| 17 | \( 1 + (3.36 + 2.38i)T \) |
| good | 5 | \( 1 + (-2.15 + 1.88i)T + (0.652 - 4.95i)T^{2} \) |
| 11 | \( 1 + (1.18 - 0.0777i)T + (10.9 - 1.43i)T^{2} \) |
| 13 | \( 1 + (2.39 + 2.39i)T + 13iT^{2} \) |
| 19 | \( 1 + (-0.848 - 6.44i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (-4.06 - 2.00i)T + (14.0 + 18.2i)T^{2} \) |
| 29 | \( 1 + (-1.95 - 9.81i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + (-2.57 + 1.27i)T + (18.8 - 24.5i)T^{2} \) |
| 37 | \( 1 + (-0.183 + 2.80i)T + (-36.6 - 4.82i)T^{2} \) |
| 41 | \( 1 + (-1.88 + 9.48i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (0.0948 + 0.229i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (-0.800 + 2.98i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (7.97 + 10.3i)T + (-13.7 + 51.1i)T^{2} \) |
| 59 | \( 1 + (-7.22 - 0.950i)T + (56.9 + 15.2i)T^{2} \) |
| 61 | \( 1 + (-0.873 - 2.57i)T + (-48.3 + 37.1i)T^{2} \) |
| 67 | \( 1 + (-9.41 - 5.43i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-11.2 - 7.52i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-9.50 - 3.22i)T + (57.9 + 44.4i)T^{2} \) |
| 79 | \( 1 + (2.45 - 4.97i)T + (-48.0 - 62.6i)T^{2} \) |
| 83 | \( 1 + (-4.57 + 11.0i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-13.6 - 3.66i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (8.40 - 1.67i)T + (89.6 - 37.1i)T^{2} \) |
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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
−10.32074698465707427545587483433, −9.503377834175180648180800198054, −8.339533464391347564622597454365, −7.35771099822492746472919848718, −6.62015409744589268501779393816, −5.31637607393283627746184104016, −5.11649661590464108539858416976, −3.67848930786614076310538776882, −2.23783799719073504174866482097, −1.04454643831224931912552304118,
2.30925627901140997610251503971, 2.83482842230770340401407870667, 4.50145107170814961648892230696, 5.15237673427705559477620616930, 6.35681418114796773651846638413, 6.55821331029485237802314194167, 8.022457167705749337353949955211, 9.213289331788112042554241329967, 9.780571432758192246336072330635, 10.87124383454444461652685295758
