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 | \( 1 + (-6.26 + 8.61i)T + (-29.9 - 92.2i)T^{2} \) |
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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
−11.34542299855305310472019051975, −10.73386779114193423273147920787, −9.724827990424295000896796406964, −9.115510274563967875275266500450, −8.407650390224868037419334069994, −7.31162471220192745802170576596, −6.14823895143341788223609198774, −4.86959620890789528109663859661, −3.05387952417620905069298510409, −1.84556192377696583715251594112,
0.36047364483587687684371299629, 1.73871719883103201366781079634, 3.89165308472777589433646660009, 5.74295857746142835325650133653, 6.68472072247106203962199017105, 7.31008645722434478569675453879, 8.349196787619787456753052021281, 8.899191801878916970091504461861, 10.21077063464351090033932209851, 10.91639498158191479704100514428