| L(s) = 1 | + (1.11 + 0.871i)2-s + (−0.264 + 1.71i)3-s + (0.482 + 1.94i)4-s + (−3.29 − 1.75i)5-s + (−1.78 + 1.67i)6-s + (−0.370 − 0.0737i)7-s + (−1.15 + 2.58i)8-s + (−2.86 − 0.904i)9-s + (−2.13 − 4.82i)10-s + (−4.94 + 4.05i)11-s + (−3.45 + 0.312i)12-s + (2.85 − 1.52i)13-s + (−0.348 − 0.405i)14-s + (3.88 − 5.17i)15-s + (−3.53 + 1.87i)16-s + (1.84 − 0.762i)17-s + ⋯ |
| L(s) = 1 | + (0.787 + 0.616i)2-s + (−0.152 + 0.988i)3-s + (0.241 + 0.970i)4-s + (−1.47 − 0.787i)5-s + (−0.728 + 0.684i)6-s + (−0.140 − 0.0278i)7-s + (−0.408 + 0.912i)8-s + (−0.953 − 0.301i)9-s + (−0.675 − 1.52i)10-s + (−1.49 + 1.22i)11-s + (−0.995 + 0.0901i)12-s + (0.791 − 0.422i)13-s + (−0.0932 − 0.108i)14-s + (1.00 − 1.33i)15-s + (−0.883 + 0.467i)16-s + (0.446 − 0.184i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.158i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0782209 - 0.983027i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0782209 - 0.983027i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.11 - 0.871i)T \) |
| 3 | \( 1 + (0.264 - 1.71i)T \) |
| good | 5 | \( 1 + (3.29 + 1.75i)T + (2.77 + 4.15i)T^{2} \) |
| 7 | \( 1 + (0.370 + 0.0737i)T + (6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (4.94 - 4.05i)T + (2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-2.85 + 1.52i)T + (7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (-1.84 + 0.762i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (1.60 - 5.28i)T + (-15.7 - 10.5i)T^{2} \) |
| 23 | \( 1 + (-6.74 - 4.50i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (1.52 + 1.24i)T + (5.65 + 28.4i)T^{2} \) |
| 31 | \( 1 + (0.401 - 0.401i)T - 31iT^{2} \) |
| 37 | \( 1 + (-1.65 - 5.44i)T + (-30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (-0.603 - 0.403i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-0.837 + 8.50i)T + (-42.1 - 8.38i)T^{2} \) |
| 47 | \( 1 + (-0.573 + 0.237i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (3.48 - 2.85i)T + (10.3 - 51.9i)T^{2} \) |
| 59 | \( 1 + (-11.0 - 5.91i)T + (32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (-0.569 - 5.78i)T + (-59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (8.13 - 0.801i)T + (65.7 - 13.0i)T^{2} \) |
| 71 | \( 1 + (3.18 + 0.633i)T + (65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (-0.359 - 1.80i)T + (-67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (1.60 - 3.87i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (0.559 - 1.84i)T + (-69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (-2.82 - 4.23i)T + (-34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (-1.32 - 1.32i)T + 97iT^{2} \) |
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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
−11.93499429800755903320241306111, −11.10461420325832082152390066309, −10.09078443143730101914243336338, −8.759896959838203912523742511791, −8.015695748334781176055313398493, −7.28997320495320471076405230892, −5.62577421593622687105271110198, −4.92777458738096670494312689744, −4.05011508859032908359771325490, −3.15977151881112787507184058348,
0.50652392360557796413042279893, 2.70557854603484745799220218113, 3.37368590302405315160212902815, 4.85679749358785871465120756072, 6.09735039449296509067760442987, 6.93699602556778719912810557072, 7.88842804160722919200842877415, 8.823466357218207631618654529740, 10.73353165379323044614069351497, 11.05634198212871327692237725246
