| L(s) = 1 | + (−2.52 + 0.514i)2-s + (2.70 + 0.439i)3-s + (4.24 − 1.81i)4-s + (0.847 + 3.43i)5-s + (−7.05 + 0.284i)6-s + (0.693 + 0.328i)7-s + (−5.54 + 3.82i)8-s + (4.28 + 1.43i)9-s + (−3.90 − 8.22i)10-s + (−1.32 − 3.95i)11-s + (12.2 − 3.03i)12-s + (−2.08 − 2.94i)13-s + (−1.91 − 0.472i)14-s + (0.781 + 9.67i)15-s + (5.60 − 5.83i)16-s + (−2.29 + 4.83i)17-s + ⋯ |
| L(s) = 1 | + (−1.78 + 0.363i)2-s + (1.56 + 0.254i)3-s + (2.12 − 0.905i)4-s + (0.378 + 1.53i)5-s + (−2.87 + 0.115i)6-s + (0.262 + 0.124i)7-s + (−1.95 + 1.35i)8-s + (1.42 + 0.477i)9-s + (−1.23 − 2.60i)10-s + (−0.398 − 1.19i)11-s + (3.55 − 0.875i)12-s + (−0.576 − 0.816i)13-s + (−0.512 − 0.126i)14-s + (0.201 + 2.49i)15-s + (1.40 − 1.45i)16-s + (−0.556 + 1.17i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.369 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.369 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.738024 + 0.500961i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.738024 + 0.500961i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (2.08 + 2.94i)T \) |
| good | 2 | \( 1 + (2.52 - 0.514i)T + (1.83 - 0.783i)T^{2} \) |
| 3 | \( 1 + (-2.70 - 0.439i)T + (2.84 + 0.950i)T^{2} \) |
| 5 | \( 1 + (-0.847 - 3.43i)T + (-4.42 + 2.32i)T^{2} \) |
| 7 | \( 1 + (-0.693 - 0.328i)T + (4.42 + 5.42i)T^{2} \) |
| 11 | \( 1 + (1.32 + 3.95i)T + (-8.79 + 6.60i)T^{2} \) |
| 17 | \( 1 + (2.29 - 4.83i)T + (-10.7 - 13.1i)T^{2} \) |
| 19 | \( 1 + (-1.08 + 0.625i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.81 + 3.14i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.242 - 1.18i)T + (-26.6 + 11.3i)T^{2} \) |
| 31 | \( 1 + (-1.61 + 3.08i)T + (-17.6 - 25.5i)T^{2} \) |
| 37 | \( 1 + (-5.46 - 8.63i)T + (-15.8 + 33.4i)T^{2} \) |
| 41 | \( 1 + (-1.11 + 6.83i)T + (-38.8 - 12.9i)T^{2} \) |
| 43 | \( 1 + (4.94 + 3.12i)T + (18.4 + 38.8i)T^{2} \) |
| 47 | \( 1 + (-1.77 + 0.215i)T + (45.6 - 11.2i)T^{2} \) |
| 53 | \( 1 + (-0.439 - 0.636i)T + (-18.7 + 49.5i)T^{2} \) |
| 59 | \( 1 + (3.29 - 3.16i)T + (2.37 - 58.9i)T^{2} \) |
| 61 | \( 1 + (-7.51 - 0.606i)T + (60.2 + 9.78i)T^{2} \) |
| 67 | \( 1 + (-3.22 + 7.57i)T + (-46.4 - 48.3i)T^{2} \) |
| 71 | \( 1 + (7.05 + 5.75i)T + (14.2 + 69.5i)T^{2} \) |
| 73 | \( 1 + (7.64 - 8.62i)T + (-8.79 - 72.4i)T^{2} \) |
| 79 | \( 1 + (-0.238 - 1.96i)T + (-76.7 + 18.9i)T^{2} \) |
| 83 | \( 1 + (-1.92 - 0.731i)T + (62.1 + 55.0i)T^{2} \) |
| 89 | \( 1 + (7.96 + 4.59i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.85 + 0.827i)T + (81.9 - 51.8i)T^{2} \) |
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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
−13.31974093151996889973393178938, −11.26582800930131086140518377187, −10.44289220405895075885572777661, −9.915687513169550550466849769448, −8.694583103312550327494451012327, −8.143981709367129058840303024707, −7.20916929314645384994084711491, −6.10721768411212323166919766461, −3.14935214784112573783235061802, −2.28805317398735887105919647157,
1.51413618322468157767354325780, 2.51272053170080896808376979212, 4.65189185393078644688745264470, 7.13977966595557185307406924661, 7.81927221090528229570266208284, 8.753941108776478581823528495350, 9.452337240846122867015363356073, 9.776550951459245628348494462943, 11.50948369861076278476728389264, 12.52939881061302229783767246956
