L(s) = 1 | + (4.33 + 2.5i)3-s + (−4.5 − 7.79i)5-s + (6.92 + i)7-s + (8.00 + 13.8i)9-s + (2.59 + 1.5i)11-s + 16·13-s − 45.0i·15-s + (3.5 − 6.06i)17-s + (9.52 − 5.5i)19-s + (27.5 + 21.6i)21-s + (−16.4 + 9.5i)23-s + (−28 + 48.4i)25-s + 35.0i·27-s − 32·29-s + (9.52 + 5.5i)31-s + ⋯ |
L(s) = 1 | + (1.44 + 0.833i)3-s + (−0.900 − 1.55i)5-s + (0.989 + 0.142i)7-s + (0.888 + 1.53i)9-s + (0.236 + 0.136i)11-s + 1.23·13-s − 3.00i·15-s + (0.205 − 0.356i)17-s + (0.501 − 0.289i)19-s + (1.30 + 1.03i)21-s + (−0.715 + 0.413i)23-s + (−1.12 + 1.93i)25-s + 1.29i·27-s − 1.10·29-s + (0.307 + 0.177i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0550i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.40982 + 0.0663793i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.40982 + 0.0663793i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-6.92 - i)T \) |
good | 3 | \( 1 + (-4.33 - 2.5i)T + (4.5 + 7.79i)T^{2} \) |
| 5 | \( 1 + (4.5 + 7.79i)T + (-12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (-2.59 - 1.5i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 16T + 169T^{2} \) |
| 17 | \( 1 + (-3.5 + 6.06i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-9.52 + 5.5i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (16.4 - 9.5i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 32T + 841T^{2} \) |
| 31 | \( 1 + (-9.52 - 5.5i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 40T + 1.68e3T^{2} \) |
| 43 | \( 1 + 40iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (73.6 - 42.5i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (3.5 - 6.06i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-45.8 - 26.5i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (39.5 + 68.4i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (9.52 + 5.5i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 48iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (71.5 - 123. i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (30.3 - 17.5i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-48.5 - 84.0i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 88T + 9.40e3T^{2} \) |
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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
−11.97576397249849419953717902285, −11.15954219697852900038961710202, −9.704674922518565832835219973942, −8.870298382664163335252915045292, −8.327002363741599132196297108391, −7.63852375603967154398646271098, −5.29450121193793909041291769494, −4.35600323221263858177462550067, −3.53908403006120739696873587713, −1.54673594754687299533832877666,
1.75067852966800252895958997178, 3.16424864408850206250656724298, 3.93120179948246574074877889344, 6.27530637637527574808436850450, 7.29455366833140844899706167295, 7.977286748951252764029864762324, 8.608851883695602818545809132685, 10.13502143819389042074502561666, 11.20133028999287884790789334806, 11.87662681532237381015156198848