L(s) = 1 | + (−1.18 + 1.18i)2-s + (0.932 + 5.11i)3-s + 5.17i·4-s + (−7.17 − 4.96i)6-s + (13.3 + 13.3i)7-s + (−15.6 − 15.6i)8-s + (−25.2 + 9.53i)9-s − 28.7i·11-s + (−26.4 + 4.83i)12-s + (−14.1 + 14.1i)13-s − 31.7·14-s − 4.25·16-s + (−18.5 + 18.5i)17-s + (18.6 − 41.3i)18-s + 49.0i·19-s + ⋯ |
L(s) = 1 | + (−0.419 + 0.419i)2-s + (0.179 + 0.983i)3-s + 0.647i·4-s + (−0.488 − 0.337i)6-s + (0.721 + 0.721i)7-s + (−0.691 − 0.691i)8-s + (−0.935 + 0.353i)9-s − 0.787i·11-s + (−0.636 + 0.116i)12-s + (−0.302 + 0.302i)13-s − 0.605·14-s − 0.0664·16-s + (−0.264 + 0.264i)17-s + (0.244 − 0.541i)18-s + 0.592i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.931 - 0.364i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.931 - 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.208197 + 1.10314i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.208197 + 1.10314i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.932 - 5.11i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (1.18 - 1.18i)T - 8iT^{2} \) |
| 7 | \( 1 + (-13.3 - 13.3i)T + 343iT^{2} \) |
| 11 | \( 1 + 28.7iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (14.1 - 14.1i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 + (18.5 - 18.5i)T - 4.91e3iT^{2} \) |
| 19 | \( 1 - 49.0iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-37.7 - 37.7i)T + 1.21e4iT^{2} \) |
| 29 | \( 1 - 125.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 247.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-127. - 127. i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 - 390. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (-39.3 + 39.3i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + (124. - 124. i)T - 1.03e5iT^{2} \) |
| 53 | \( 1 + (-160. - 160. i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 + 729.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 2T + 2.26e5T^{2} \) |
| 67 | \( 1 + (-329. - 329. i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 + 171. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-279. + 279. i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 - 48.0iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (144. + 144. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 - 1.41e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (908. + 908. i)T + 9.12e5iT^{2} \) |
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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
−14.83308077814925462065484306755, −13.65140290484601497752852215921, −12.09890181322226665237453757289, −11.23040481911644280142956287305, −9.776400365778931262893402581715, −8.667016772988260509748904231887, −8.025231426049988264520307317857, −6.15650483179051940131039844169, −4.58537515729170967344608694103, −2.98666347436891069574896718930,
0.835816795275370201792001187246, 2.35456002458883041322480859123, 4.91288570277976723812451043895, 6.56613746938243289592000115940, 7.74406648019223650536074587681, 8.990449943206975323959456608043, 10.29751043100024111105049185130, 11.31349348280292438982109773423, 12.34242169013754562911415349532, 13.66162208762178973938185557699