| L(s) = 1 | + (−1.24 + 0.679i)2-s + (1.09 − 1.34i)3-s + (1.07 − 1.68i)4-s + (0.708 + 0.378i)5-s + (−0.442 + 2.40i)6-s + (−4.90 − 0.974i)7-s + (−0.186 + 2.82i)8-s + (−0.609 − 2.93i)9-s + (−1.13 + 0.0121i)10-s + (3.64 − 2.99i)11-s + (−1.09 − 3.28i)12-s + (1.28 − 0.687i)13-s + (6.74 − 2.12i)14-s + (1.28 − 0.537i)15-s + (−1.68 − 3.62i)16-s + (−4.17 + 1.72i)17-s + ⋯ |
| L(s) = 1 | + (−0.876 + 0.480i)2-s + (0.631 − 0.775i)3-s + (0.537 − 0.843i)4-s + (0.316 + 0.169i)5-s + (−0.180 + 0.983i)6-s + (−1.85 − 0.368i)7-s + (−0.0660 + 0.997i)8-s + (−0.203 − 0.979i)9-s + (−0.359 + 0.00384i)10-s + (1.09 − 0.901i)11-s + (−0.314 − 0.949i)12-s + (0.356 − 0.190i)13-s + (1.80 − 0.567i)14-s + (0.331 − 0.138i)15-s + (−0.421 − 0.906i)16-s + (−1.01 + 0.419i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0681 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0681 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.579976 - 0.620931i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.579976 - 0.620931i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.24 - 0.679i)T \) |
| 3 | \( 1 + (-1.09 + 1.34i)T \) |
| good | 5 | \( 1 + (-0.708 - 0.378i)T + (2.77 + 4.15i)T^{2} \) |
| 7 | \( 1 + (4.90 + 0.974i)T + (6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (-3.64 + 2.99i)T + (2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-1.28 + 0.687i)T + (7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (4.17 - 1.72i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-1.43 + 4.72i)T + (-15.7 - 10.5i)T^{2} \) |
| 23 | \( 1 + (3.44 + 2.30i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (-6.17 - 5.06i)T + (5.65 + 28.4i)T^{2} \) |
| 31 | \( 1 + (1.73 - 1.73i)T - 31iT^{2} \) |
| 37 | \( 1 + (1.79 + 5.92i)T + (-30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (-1.22 - 0.815i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-0.444 + 4.51i)T + (-42.1 - 8.38i)T^{2} \) |
| 47 | \( 1 + (-1.80 + 0.747i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (0.0938 - 0.0770i)T + (10.3 - 51.9i)T^{2} \) |
| 59 | \( 1 + (-1.86 - 0.995i)T + (32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (-0.477 - 4.84i)T + (-59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (-12.5 + 1.23i)T + (65.7 - 13.0i)T^{2} \) |
| 71 | \( 1 + (-0.718 - 0.142i)T + (65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (-1.98 - 9.98i)T + (-67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (-2.91 + 7.02i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (0.838 - 2.76i)T + (-69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (8.43 + 12.6i)T + (-34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (-10.1 - 10.1i)T + 97iT^{2} \) |
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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
−10.84124062843505033425456039707, −9.915638043749575406747815785937, −9.011264290012850940901010440577, −8.603750822577945442326670395906, −7.13104325600587679392049474727, −6.53497809955258430180438296578, −6.03898773194132858143439391340, −3.68887678691419535941549481152, −2.49064010561324555455057230713, −0.67859694306992880311820543751,
2.06370093207395165143453881160, 3.31501276366774274046117731222, 4.15681203542970171043707571690, 6.07209443173973235122839619355, 6.96030763311699372196610411206, 8.241107730635456877264135436955, 9.326417432990362330173620680317, 9.571781960204191780240268177509, 10.16643500917574713553184560896, 11.47676071168818867190952996022
