L(s) = 1 | − 9.63·2-s + (15.3 + 2.74i)3-s + 60.8·4-s + 48.0i·5-s + (−147. − 26.4i)6-s + 55.2·7-s − 278.·8-s + (227. + 84.1i)9-s − 463. i·10-s − 33.0·11-s + (934. + 166. i)12-s + 847. i·13-s − 532.·14-s + (−131. + 738. i)15-s + 733.·16-s + 1.12e3i·17-s + ⋯ |
L(s) = 1 | − 1.70·2-s + (0.984 + 0.175i)3-s + 1.90·4-s + 0.860i·5-s + (−1.67 − 0.299i)6-s + 0.426·7-s − 1.53·8-s + (0.938 + 0.346i)9-s − 1.46i·10-s − 0.0822·11-s + (1.87 + 0.334i)12-s + 1.39i·13-s − 0.726·14-s + (−0.151 + 0.847i)15-s + 0.716·16-s + 0.947i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.610 - 0.791i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.610 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.053148883\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.053148883\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-15.3 - 2.74i)T \) |
| 59 | \( 1 + (-1.98e4 - 1.79e4i)T \) |
good | 2 | \( 1 + 9.63T + 32T^{2} \) |
| 5 | \( 1 - 48.0iT - 3.12e3T^{2} \) |
| 7 | \( 1 - 55.2T + 1.68e4T^{2} \) |
| 11 | \( 1 + 33.0T + 1.61e5T^{2} \) |
| 13 | \( 1 - 847. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.12e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.38e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 468.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 408. iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 5.08e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 1.57e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.69e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 1.11e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.20e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.98e3iT - 4.18e8T^{2} \) |
| 61 | \( 1 - 5.04e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 3.40e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 1.89e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + 2.36e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 6.19e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.15e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.23e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.50e4iT - 8.58e9T^{2} \) |
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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
−11.61753667092150694027289818630, −10.73730085596997742672183257301, −10.01788306948471100513156684880, −9.007001522820624806198485839104, −8.325624740704434795398159008068, −7.32054653440315918777609181878, −6.51468995900207134484807016914, −4.14619406342332140357351691721, −2.51608794195458158303961337937, −1.59787117796514017933687308803,
0.52427292336542579293275765615, 1.59590029687083066169891730548, 2.94389101267618124936122250004, 4.90013397890523030530581895154, 6.76428993208376641463793726877, 7.905984407205892034553438599046, 8.377669958441402230646018771089, 9.217990132139836337375931533268, 10.07885635287565291451993245379, 11.05545353862231846235134706705