L(s) = 1 | + (0.707 + 0.707i)2-s + (0.292 + 1.70i)3-s + 1.00i·4-s + (−0.999 + 1.41i)6-s + (1 − i)7-s + (−0.707 + 0.707i)8-s + (−2.82 + i)9-s − 1.41i·11-s + (−1.70 + 0.292i)12-s + 1.41·14-s − 1.00·16-s + (1.41 + 1.41i)17-s + (−2.70 − 1.29i)18-s − 4i·19-s + (2 + 1.41i)21-s + (1.00 − 1.00i)22-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (0.169 + 0.985i)3-s + 0.500i·4-s + (−0.408 + 0.577i)6-s + (0.377 − 0.377i)7-s + (−0.250 + 0.250i)8-s + (−0.942 + 0.333i)9-s − 0.426i·11-s + (−0.492 + 0.0845i)12-s + 0.377·14-s − 0.250·16-s + (0.342 + 0.342i)17-s + (−0.638 − 0.304i)18-s − 0.917i·19-s + (0.436 + 0.308i)21-s + (0.213 − 0.213i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0618 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0618 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.07291 + 1.00846i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.07291 + 1.00846i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 + (-0.292 - 1.70i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-1 + i)T - 7iT^{2} \) |
| 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 + 13iT^{2} \) |
| 17 | \( 1 + (-1.41 - 1.41i)T + 17iT^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + (-2.82 + 2.82i)T - 23iT^{2} \) |
| 29 | \( 1 - 7.07T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + (6 - 6i)T - 37iT^{2} \) |
| 41 | \( 1 + 5.65iT - 41T^{2} \) |
| 43 | \( 1 + (6 + 6i)T + 43iT^{2} \) |
| 47 | \( 1 + 47iT^{2} \) |
| 53 | \( 1 + (2.82 - 2.82i)T - 53iT^{2} \) |
| 59 | \( 1 + 9.89T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + (-4 + 4i)T - 67iT^{2} \) |
| 71 | \( 1 - 14.1iT - 71T^{2} \) |
| 73 | \( 1 + (-5 - 5i)T + 73iT^{2} \) |
| 79 | \( 1 + 6iT - 79T^{2} \) |
| 83 | \( 1 + (8.48 - 8.48i)T - 83iT^{2} \) |
| 89 | \( 1 + 2.82T + 89T^{2} \) |
| 97 | \( 1 + (3 - 3i)T - 97iT^{2} \) |
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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
−13.64689751035716128818865256128, −12.29516435834710054854808524304, −11.15083634974028438125166343030, −10.32989234836431187015106874474, −8.985692380732752015401268507962, −8.142243146865085970514718100456, −6.73482976201572640241325291537, −5.34819308263600551574721031375, −4.38952672316284747988970119318, −3.07132677816928570787902344062,
1.69579203176320333609197701026, 3.19366268773046050282275955484, 5.00830344452743109570541413242, 6.20138554022249356674161500386, 7.43143313213841432231629025837, 8.549842992853152009203692468559, 9.777141402896229544445029296531, 11.10201394673275814285660561685, 12.03345843458782057860913525056, 12.62801705934496964330163603216