L(s) = 1 | + (−5.20 + 2.22i)2-s + (−15.5 − 0.545i)3-s + (22.1 − 23.1i)4-s + (−17.3 − 53.1i)5-s + (82.2 − 31.7i)6-s + 106.·7-s + (−63.7 + 169. i)8-s + (242. + 16.9i)9-s + (208. + 238. i)10-s − 278.·11-s + (−357. + 348. i)12-s − 699. i·13-s + (−552. + 235. i)14-s + (240. + 837. i)15-s + (−44.7 − 1.02e3i)16-s − 1.73e3·17-s + ⋯ |
L(s) = 1 | + (−0.919 + 0.392i)2-s + (−0.999 − 0.0349i)3-s + (0.691 − 0.722i)4-s + (−0.309 − 0.950i)5-s + (0.932 − 0.360i)6-s + 0.819·7-s + (−0.352 + 0.935i)8-s + (0.997 + 0.0699i)9-s + (0.658 + 0.752i)10-s − 0.692·11-s + (−0.716 + 0.697i)12-s − 1.14i·13-s + (−0.753 + 0.321i)14-s + (0.276 + 0.961i)15-s + (−0.0436 − 0.999i)16-s − 1.45·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.885 - 0.464i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.885 - 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.0205559 + 0.0833588i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0205559 + 0.0833588i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (5.20 - 2.22i)T \) |
| 3 | \( 1 + (15.5 + 0.545i)T \) |
| 5 | \( 1 + (17.3 + 53.1i)T \) |
good | 7 | \( 1 - 106.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 278.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 699. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.73e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.22e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 1.75e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 2.52e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 5.69e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 977. iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.74e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 5.03e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 4.80e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.72e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.28e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.22e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.79e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.01e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.31e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 8.40e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 5.67e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 1.19e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 2.38e4iT - 8.58e9T^{2} \) |
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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
−15.11011959851062884366655624650, −13.24471601982847149718807519023, −12.01642891478042661626682474389, −11.02711549138277522261399374040, −10.01885518660300403461094755320, −8.452470968755405093520675453770, −7.57033829961711580855147672946, −5.86086208254448184387873060693, −4.83812093976844945484964194383, −1.44379136926987651960106180900,
0.06360614824055522619761376413, 2.20436468094544870059300719913, 4.42996566495210348937190796661, 6.54576175389371611533105086261, 7.41567224196215487130662076952, 8.999045952605520595636149754935, 10.54780688105621284023833742423, 11.13577057969053694534577805560, 11.87802070796753414172581752171, 13.38284428352063650417269298054