L(s) = 1 | + 10.4i·2-s − 77.3·4-s + 44.1·5-s + 173. i·7-s − 474. i·8-s + 461. i·10-s + 517.·11-s − 1.13e3·13-s − 1.81e3·14-s + 2.48e3·16-s − 1.73e3·17-s − 2.75e3i·19-s − 3.41e3·20-s + 5.41e3i·22-s + (−2.39e3 + 843. i)23-s + ⋯ |
L(s) = 1 | + 1.84i·2-s − 2.41·4-s + 0.789·5-s + 1.33i·7-s − 2.62i·8-s + 1.45i·10-s + 1.29·11-s − 1.85·13-s − 2.47·14-s + 2.43·16-s − 1.45·17-s − 1.74i·19-s − 1.90·20-s + 2.38i·22-s + (−0.943 + 0.332i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 + 0.578i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.1808325648\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1808325648\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 + (2.39e3 - 843. i)T \) |
good | 2 | \( 1 - 10.4iT - 32T^{2} \) |
| 5 | \( 1 - 44.1T + 3.12e3T^{2} \) |
| 7 | \( 1 - 173. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 517.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.13e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.73e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.75e3iT - 2.47e6T^{2} \) |
| 29 | \( 1 - 3.33e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 6.81e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.31e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 8.22e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 5.22e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.03e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 1.97e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.80e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 1.11e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 5.00e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 4.11e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 5.10e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.86e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 3.78e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.83e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.18e4iT - 8.58e9T^{2} \) |
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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
−12.67606229059607240763029368767, −11.66903172235321295228231172688, −9.678091202908666037510810687887, −9.242704399766762338088135784774, −8.420662537289912025637118346463, −6.98844502756214177356375347960, −6.41462619865281698127003426916, −5.33568048294940958829965703923, −4.53686025380928369738027173535, −2.32403107092891092725951493564,
0.05397492453066080792027813398, 1.41868019056716282498898402036, 2.37934204193813803632023235337, 3.94519253622554481769067888515, 4.57077749132055926910494259352, 6.32284819726006762479240040832, 7.85740488777200414961045833652, 9.308884316048769724846872710993, 9.940541907052063873885289075608, 10.48520944374323897485327976872