L(s) = 1 | + (−0.292 − 0.292i)2-s + (−1.41 + i)3-s − 1.82i·4-s + (0.707 + 0.121i)6-s + (3.41 − 3.41i)7-s + (−1.12 + 1.12i)8-s + (1.00 − 2.82i)9-s − i·11-s + (1.82 + 2.58i)12-s + (2 + 2i)13-s − 2·14-s − 3·16-s + (−2.82 − 2.82i)17-s + (−1.12 + 0.535i)18-s + 2.82i·19-s + ⋯ |
L(s) = 1 | + (−0.207 − 0.207i)2-s + (−0.816 + 0.577i)3-s − 0.914i·4-s + (0.288 + 0.0495i)6-s + (1.29 − 1.29i)7-s + (−0.396 + 0.396i)8-s + (0.333 − 0.942i)9-s − 0.301i·11-s + (0.527 + 0.746i)12-s + (0.554 + 0.554i)13-s − 0.534·14-s − 0.750·16-s + (−0.685 − 0.685i)17-s + (−0.264 + 0.126i)18-s + 0.648i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.559245 - 0.845217i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.559245 - 0.845217i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.41 - i)T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + iT \) |
good | 2 | \( 1 + (0.292 + 0.292i)T + 2iT^{2} \) |
| 7 | \( 1 + (-3.41 + 3.41i)T - 7iT^{2} \) |
| 13 | \( 1 + (-2 - 2i)T + 13iT^{2} \) |
| 17 | \( 1 + (2.82 + 2.82i)T + 17iT^{2} \) |
| 19 | \( 1 - 2.82iT - 19T^{2} \) |
| 23 | \( 1 + (-3.24 + 3.24i)T - 23iT^{2} \) |
| 29 | \( 1 + 0.828T + 29T^{2} \) |
| 31 | \( 1 + 3.17T + 31T^{2} \) |
| 37 | \( 1 + (-0.171 + 0.171i)T - 37iT^{2} \) |
| 41 | \( 1 + 7.65iT - 41T^{2} \) |
| 43 | \( 1 + (0.242 + 0.242i)T + 43iT^{2} \) |
| 47 | \( 1 + (7.24 + 7.24i)T + 47iT^{2} \) |
| 53 | \( 1 + (-1 + i)T - 53iT^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 6.48T + 61T^{2} \) |
| 67 | \( 1 + (-6.41 + 6.41i)T - 67iT^{2} \) |
| 71 | \( 1 + 2.48iT - 71T^{2} \) |
| 73 | \( 1 + 73iT^{2} \) |
| 79 | \( 1 + 10.8iT - 79T^{2} \) |
| 83 | \( 1 + (9.07 - 9.07i)T - 83iT^{2} \) |
| 89 | \( 1 - 9.65T + 89T^{2} \) |
| 97 | \( 1 + (10.6 - 10.6i)T - 97iT^{2} \) |
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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
−10.25835523307782678635141197685, −9.293653019480754007015969552535, −8.478218723189264909730968647208, −7.18585303658935740544432848513, −6.44633939726195695453190438755, −5.32669152771374113443053634103, −4.68714575699672639251431252438, −3.80131330969066967900311546767, −1.75735816350239481435046658463, −0.62600548018974399612658021614,
1.63304861552945845191029180550, 2.79307672292130502550070732108, 4.40573890163771643283712766748, 5.27950012327672961362197637025, 6.16731434413030496508967783157, 7.13500561494779272224453310191, 8.021791799718282490655185084172, 8.501022986552727824103483335986, 9.415371067603548161626548408080, 10.91425113505337666255889579067