| L(s) = 1 | + (−0.988 + 0.149i)2-s + (−1.39 − 0.949i)3-s + (0.955 − 0.294i)4-s + (−0.169 + 2.26i)5-s + (1.51 + 0.731i)6-s + (−0.900 + 0.433i)8-s + (−0.0586 − 0.149i)9-s + (−0.169 − 2.26i)10-s + (0.704 − 1.79i)11-s + (−1.61 − 0.496i)12-s + (−0.691 − 0.866i)13-s + (2.38 − 2.99i)15-s + (0.826 − 0.563i)16-s + (2.34 − 2.17i)17-s + (0.0802 + 0.138i)18-s + (−3.66 + 6.35i)19-s + ⋯ |
| L(s) = 1 | + (−0.699 + 0.105i)2-s + (−0.803 − 0.548i)3-s + (0.477 − 0.147i)4-s + (−0.0759 + 1.01i)5-s + (0.619 + 0.298i)6-s + (−0.318 + 0.153i)8-s + (−0.0195 − 0.0497i)9-s + (−0.0536 − 0.716i)10-s + (0.212 − 0.541i)11-s + (−0.464 − 0.143i)12-s + (−0.191 − 0.240i)13-s + (0.616 − 0.772i)15-s + (0.206 − 0.140i)16-s + (0.568 − 0.527i)17-s + (0.0189 + 0.0327i)18-s + (−0.841 + 1.45i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 686 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.186i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 686 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.749330 + 0.0704241i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.749330 + 0.0704241i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.988 - 0.149i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (1.39 + 0.949i)T + (1.09 + 2.79i)T^{2} \) |
| 5 | \( 1 + (0.169 - 2.26i)T + (-4.94 - 0.745i)T^{2} \) |
| 11 | \( 1 + (-0.704 + 1.79i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (0.691 + 0.866i)T + (-2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (-2.34 + 2.17i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (3.66 - 6.35i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.79 - 2.59i)T + (1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (1.59 + 7.00i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (-2.16 - 3.74i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-8.01 - 2.47i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (-4.54 + 2.18i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (0.813 + 0.391i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (-11.9 + 1.80i)T + (44.9 - 13.8i)T^{2} \) |
| 53 | \( 1 + (-13.5 + 4.17i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (-0.0132 - 0.176i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (6.13 + 1.89i)T + (50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (-7.26 - 12.5i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.21 - 9.70i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-3.67 - 0.554i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (-4.25 + 7.36i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.671 + 0.842i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-1.53 - 3.92i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 - 5.62T + 97T^{2} \) |
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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
−10.53118754516532021300172965570, −9.821118885488914502445369860714, −8.720837906541578193454672748054, −7.70330431668974360875276979160, −7.01251704064118458180711283504, −6.16513838291312218829271772948, −5.57929209744664366424104840841, −3.77018178721358836349579896115, −2.56847250621077567123855955478, −0.921510967480244455826724794281,
0.793083527990611512520581449021, 2.42045729106366545724080764300, 4.24201685351729931957291214369, 4.89845989457705448251884907035, 5.91702357082426807056528430485, 6.97970870412592335863272590481, 8.010524844059452073049802091521, 8.970033934346327894063873974765, 9.461108410937080918916263883424, 10.63674921357003379516121768082
