L(s) = 1 | + (−2.32 + 0.754i)2-s + (−0.587 − 0.809i)3-s + (3.19 − 2.32i)4-s + (1.97 + 1.43i)6-s − 3.44i·7-s + (−2.80 + 3.85i)8-s + (−0.309 + 0.951i)9-s + (−1.00 − 3.10i)11-s + (−3.76 − 1.22i)12-s + (3.07 + 0.998i)13-s + (2.59 + 7.98i)14-s + (1.15 − 3.54i)16-s + (−2.97 + 4.08i)17-s − 2.44i·18-s + (−2.49 − 1.81i)19-s + ⋯ |
L(s) = 1 | + (−1.64 + 0.533i)2-s + (−0.339 − 0.467i)3-s + (1.59 − 1.16i)4-s + (0.805 + 0.585i)6-s − 1.30i·7-s + (−0.991 + 1.36i)8-s + (−0.103 + 0.317i)9-s + (−0.304 − 0.936i)11-s + (−1.08 − 0.352i)12-s + (0.852 + 0.277i)13-s + (0.693 + 2.13i)14-s + (0.288 − 0.887i)16-s + (−0.720 + 0.991i)17-s − 0.575i·18-s + (−0.571 − 0.415i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.646 + 0.762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.646 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.129314 - 0.279306i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.129314 - 0.279306i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.587 + 0.809i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (2.32 - 0.754i)T + (1.61 - 1.17i)T^{2} \) |
| 7 | \( 1 + 3.44iT - 7T^{2} \) |
| 11 | \( 1 + (1.00 + 3.10i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-3.07 - 0.998i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (2.97 - 4.08i)T + (-5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (2.49 + 1.81i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.47 + 0.478i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (2.52 - 1.83i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (6.02 + 4.37i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (5.47 + 1.77i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.67 + 5.15i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 2.53iT - 43T^{2} \) |
| 47 | \( 1 + (4.15 + 5.72i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (5.96 + 8.21i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (0.534 - 1.64i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-2.42 - 7.45i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (1.08 - 1.49i)T + (-20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-0.577 + 0.419i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (1.78 - 0.581i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (10.7 - 7.83i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (2.32 - 3.20i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (2.63 + 8.11i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-6.26 - 8.61i)T + (-29.9 + 92.2i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.91639498158191479704100514428, −10.21077063464351090033932209851, −8.899191801878916970091504461861, −8.349196787619787456753052021281, −7.31008645722434478569675453879, −6.68472072247106203962199017105, −5.74295857746142835325650133653, −3.89165308472777589433646660009, −1.73871719883103201366781079634, −0.36047364483587687684371299629,
1.84556192377696583715251594112, 3.05387952417620905069298510409, 4.86959620890789528109663859661, 6.14823895143341788223609198774, 7.31162471220192745802170576596, 8.407650390224868037419334069994, 9.115510274563967875275266500450, 9.724827990424295000896796406964, 10.73386779114193423273147920787, 11.34542299855305310472019051975