| L(s) = 1 | + (−2.36 − 0.177i)2-s + (0.532 − 2.33i)3-s + (3.58 + 0.540i)4-s + (0.329 + 1.06i)5-s + (−1.67 + 5.42i)6-s + (1.40 − 2.24i)7-s + (−3.76 − 0.859i)8-s + (−2.45 − 1.18i)9-s + (−0.590 − 2.58i)10-s + (1.47 + 3.05i)11-s + (3.17 − 8.08i)12-s + (−3.11 − 1.82i)13-s + (−3.72 + 5.05i)14-s + (2.66 − 0.199i)15-s + (1.82 + 0.562i)16-s + (0.844 − 2.15i)17-s + ⋯ |
| L(s) = 1 | + (−1.67 − 0.125i)2-s + (0.307 − 1.34i)3-s + (1.79 + 0.270i)4-s + (0.147 + 0.477i)5-s + (−0.683 + 2.21i)6-s + (0.532 − 0.846i)7-s + (−1.33 − 0.303i)8-s + (−0.818 − 0.394i)9-s + (−0.186 − 0.817i)10-s + (0.443 + 0.921i)11-s + (0.915 − 2.33i)12-s + (−0.863 − 0.505i)13-s + (−0.996 + 1.34i)14-s + (0.688 − 0.0516i)15-s + (0.456 + 0.140i)16-s + (0.204 − 0.521i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.629 + 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.629 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.317000 - 0.665066i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.317000 - 0.665066i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-1.40 + 2.24i)T \) |
| 13 | \( 1 + (3.11 + 1.82i)T \) |
| good | 2 | \( 1 + (2.36 + 0.177i)T + (1.97 + 0.298i)T^{2} \) |
| 3 | \( 1 + (-0.532 + 2.33i)T + (-2.70 - 1.30i)T^{2} \) |
| 5 | \( 1 + (-0.329 - 1.06i)T + (-4.13 + 2.81i)T^{2} \) |
| 11 | \( 1 + (-1.47 - 3.05i)T + (-6.85 + 8.60i)T^{2} \) |
| 17 | \( 1 + (-0.844 + 2.15i)T + (-12.4 - 11.5i)T^{2} \) |
| 19 | \( 1 + 1.22iT - 19T^{2} \) |
| 23 | \( 1 + (1.36 + 3.48i)T + (-16.8 + 15.6i)T^{2} \) |
| 29 | \( 1 + (-0.429 + 1.09i)T + (-21.2 - 19.7i)T^{2} \) |
| 31 | \( 1 + (-7.21 - 4.16i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.06 + 7.06i)T + (-35.3 + 10.9i)T^{2} \) |
| 41 | \( 1 + (0.0827 + 0.268i)T + (-33.8 + 23.0i)T^{2} \) |
| 43 | \( 1 + (8.44 + 2.60i)T + (35.5 + 24.2i)T^{2} \) |
| 47 | \( 1 + (0.702 - 1.02i)T + (-17.1 - 43.7i)T^{2} \) |
| 53 | \( 1 + (-2.12 - 5.41i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (0.332 + 0.358i)T + (-4.40 + 58.8i)T^{2} \) |
| 61 | \( 1 + (4.04 + 5.07i)T + (-13.5 + 59.4i)T^{2} \) |
| 67 | \( 1 + 13.3iT - 67T^{2} \) |
| 71 | \( 1 + (2.63 - 1.03i)T + (52.0 - 48.2i)T^{2} \) |
| 73 | \( 1 + (0.772 + 1.13i)T + (-26.6 + 67.9i)T^{2} \) |
| 79 | \( 1 + (-4.37 - 7.58i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.46 - 3.04i)T + (-51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (-5.04 - 7.40i)T + (-32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + (-8.94 - 5.16i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.23354059292523284727489304834, −9.389070723627623247983864658531, −8.273326298096821843879340672055, −7.74273293334553246105047405418, −6.95956521605503465620220739962, −6.65880204497924090979010215753, −4.71769437436674798827618043113, −2.74289623046842130622557680598, −1.85582580807134000809605144233, −0.70604409407717312612563176703,
1.52525267029622145298889439762, 2.98300475944543503919572212262, 4.43409548037274909723867864672, 5.45530913006850842205955554062, 6.60098926319274173777663432784, 7.965164023957231740917190207174, 8.584301342189747269036790222662, 9.152327109679487663538100222801, 9.815739641671001852163111946613, 10.42969863029522526211006067389
