| 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
−10.63674921357003379516121768082, −9.461108410937080918916263883424, −8.970033934346327894063873974765, −8.010524844059452073049802091521, −6.97970870412592335863272590481, −5.91702357082426807056528430485, −4.89845989457705448251884907035, −4.24201685351729931957291214369, −2.42045729106366545724080764300, −0.793083527990611512520581449021,
0.921510967480244455826724794281, 2.56847250621077567123855955478, 3.77018178721358836349579896115, 5.57929209744664366424104840841, 6.16513838291312218829271772948, 7.01251704064118458180711283504, 7.70330431668974360875276979160, 8.720837906541578193454672748054, 9.821118885488914502445369860714, 10.53118754516532021300172965570
