L(s) = 1 | + (−1.70 − 2.25i)2-s + (3.86 + 3.47i)3-s + (−2.19 + 7.69i)4-s + (−13.5 + 13.5i)5-s + (1.27 − 14.6i)6-s − 19.7·7-s + (21.1 − 8.14i)8-s + (2.80 + 26.8i)9-s + (53.7 + 7.52i)10-s + (20.5 + 20.5i)11-s + (−35.2 + 22.0i)12-s + (36.7 − 36.7i)13-s + (33.6 + 44.5i)14-s + (−99.6 + 5.19i)15-s + (−54.3 − 33.7i)16-s − 4.20i·17-s + ⋯ |
L(s) = 1 | + (−0.602 − 0.798i)2-s + (0.742 + 0.669i)3-s + (−0.274 + 0.961i)4-s + (−1.21 + 1.21i)5-s + (0.0868 − 0.996i)6-s − 1.06·7-s + (0.932 − 0.360i)8-s + (0.103 + 0.994i)9-s + (1.70 + 0.238i)10-s + (0.562 + 0.562i)11-s + (−0.847 + 0.530i)12-s + (0.783 − 0.783i)13-s + (0.641 + 0.850i)14-s + (−1.71 + 0.0893i)15-s + (−0.849 − 0.527i)16-s − 0.0599i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.111 - 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.111 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.615522 + 0.550507i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.615522 + 0.550507i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.70 + 2.25i)T \) |
| 3 | \( 1 + (-3.86 - 3.47i)T \) |
good | 5 | \( 1 + (13.5 - 13.5i)T - 125iT^{2} \) |
| 7 | \( 1 + 19.7T + 343T^{2} \) |
| 11 | \( 1 + (-20.5 - 20.5i)T + 1.33e3iT^{2} \) |
| 13 | \( 1 + (-36.7 + 36.7i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 + 4.20iT - 4.91e3T^{2} \) |
| 19 | \( 1 + (-38.6 - 38.6i)T + 6.85e3iT^{2} \) |
| 23 | \( 1 - 69.4iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (23.0 + 23.0i)T + 2.43e4iT^{2} \) |
| 31 | \( 1 - 219. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (-68.0 - 68.0i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 - 325.T + 6.89e4T^{2} \) |
| 43 | \( 1 + (36.9 - 36.9i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + 192.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-461. + 461. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (0.977 + 0.977i)T + 2.05e5iT^{2} \) |
| 61 | \( 1 + (-216. + 216. i)T - 2.26e5iT^{2} \) |
| 67 | \( 1 + (27.8 + 27.8i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 - 786. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 510. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 230. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (-593. + 593. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 + 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 805.T + 9.12e5T^{2} \) |
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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
−15.59819843582563273538395161377, −14.39159346433814945150015685739, −12.98700949369745058546770514978, −11.61536378058275378610517720851, −10.54917833049926492937717929334, −9.654086517756231731218794028683, −8.232764649513817741730540688531, −7.09391421501882621586174162977, −3.82936744973260962485388808328, −3.08399852654346852131349676802,
0.75028334807162743170730507797, 4.02107790482526373328137834442, 6.27294838704499836768239416371, 7.57043996112636832060797400702, 8.720513620590267450459109769125, 9.311657116561932115386732995179, 11.51341194191391939452278663551, 12.82287167619856717217437945970, 13.78681553613599102508041325391, 15.18036004862406583336330752321