| L(s) = 1 | − 1.70i·2-s − 0.903·4-s + (−1.83 + 1.27i)5-s + 0.338i·7-s − 1.86i·8-s + (2.17 + 3.12i)10-s − 1.26·11-s + 6.57i·13-s + 0.576·14-s − 4.99·16-s + 3.65i·17-s − 19-s + (1.65 − 1.15i)20-s + 2.14i·22-s + 5.14i·23-s + ⋯ |
| L(s) = 1 | − 1.20i·2-s − 0.451·4-s + (−0.820 + 0.571i)5-s + 0.127i·7-s − 0.660i·8-s + (0.688 + 0.988i)10-s − 0.380·11-s + 1.82i·13-s + 0.154·14-s − 1.24·16-s + 0.887i·17-s − 0.229·19-s + (0.370 − 0.258i)20-s + 0.458i·22-s + 1.07i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 - 0.571i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.820 - 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.839394 + 0.263497i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.839394 + 0.263497i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.83 - 1.27i)T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 1.70iT - 2T^{2} \) |
| 7 | \( 1 - 0.338iT - 7T^{2} \) |
| 11 | \( 1 + 1.26T + 11T^{2} \) |
| 13 | \( 1 - 6.57iT - 13T^{2} \) |
| 17 | \( 1 - 3.65iT - 17T^{2} \) |
| 23 | \( 1 - 5.14iT - 23T^{2} \) |
| 29 | \( 1 + 4.73T + 29T^{2} \) |
| 31 | \( 1 + 3.25T + 31T^{2} \) |
| 37 | \( 1 + 4.44iT - 37T^{2} \) |
| 41 | \( 1 - 4.93T + 41T^{2} \) |
| 43 | \( 1 - 9.32iT - 43T^{2} \) |
| 47 | \( 1 - 8.42iT - 47T^{2} \) |
| 53 | \( 1 - 6.98iT - 53T^{2} \) |
| 59 | \( 1 - 2.34T + 59T^{2} \) |
| 61 | \( 1 - 14.2T + 61T^{2} \) |
| 67 | \( 1 - 0.160iT - 67T^{2} \) |
| 71 | \( 1 + 0.160T + 71T^{2} \) |
| 73 | \( 1 + 5.07iT - 73T^{2} \) |
| 79 | \( 1 + 16.6T + 79T^{2} \) |
| 83 | \( 1 + 6.81iT - 83T^{2} \) |
| 89 | \( 1 + 16.3T + 89T^{2} \) |
| 97 | \( 1 + 13.7iT - 97T^{2} \) |
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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
−10.51344466022249064102587198599, −9.575133156982438846716772320388, −8.875101690855909627154781642424, −7.64438190369530883268991291704, −6.95356951386927737738948557211, −5.94748762891868390105496051198, −4.31984044298076593351723323024, −3.81961890652419412619883200494, −2.66834768524640797283160961611, −1.62199580226324057033006035322,
0.41881261991320426040993513060, 2.61444743923774402578148143393, 3.93629005660769089437542120368, 5.19181315801253850650202195475, 5.52845519766881621765078129622, 6.89609144890428453491535221463, 7.49446229387208559228250268289, 8.274542368130236049073030164222, 8.750626917917287296198677316823, 10.04383419604178040816580083601
