| L(s) = 1 | + (3.50 − 2.02i)2-s + (0.218 − 5.19i)3-s + (4.20 − 7.28i)4-s + (−18.5 + 4.97i)5-s + (−9.75 − 18.6i)6-s + (−10.4 − 2.80i)7-s − 1.69i·8-s + (−26.9 − 2.26i)9-s + (−55.0 + 55.0i)10-s + (−22.6 − 6.05i)11-s + (−36.9 − 23.4i)12-s + (39.3 − 68.1i)13-s + (−42.4 + 11.3i)14-s + (21.7 + 97.3i)15-s + (30.2 + 52.3i)16-s + (18.6 − 67.5i)17-s + ⋯ |
| L(s) = 1 | + (1.24 − 0.716i)2-s + (0.0420 − 0.999i)3-s + (0.526 − 0.911i)4-s + (−1.65 + 0.444i)5-s + (−0.663 − 1.26i)6-s + (−0.566 − 0.151i)7-s − 0.0748i·8-s + (−0.996 − 0.0840i)9-s + (−1.73 + 1.73i)10-s + (−0.619 − 0.166i)11-s + (−0.888 − 0.563i)12-s + (0.839 − 1.45i)13-s + (−0.810 + 0.217i)14-s + (0.374 + 1.67i)15-s + (0.472 + 0.818i)16-s + (0.266 − 0.963i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.313i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.217311 + 1.35352i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.217311 + 1.35352i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.218 + 5.19i)T \) |
| 17 | \( 1 + (-18.6 + 67.5i)T \) |
| good | 2 | \( 1 + (-3.50 + 2.02i)T + (4 - 6.92i)T^{2} \) |
| 5 | \( 1 + (18.5 - 4.97i)T + (108. - 62.5i)T^{2} \) |
| 7 | \( 1 + (10.4 + 2.80i)T + (297. + 171.5i)T^{2} \) |
| 11 | \( 1 + (22.6 + 6.05i)T + (1.15e3 + 665.5i)T^{2} \) |
| 13 | \( 1 + (-39.3 + 68.1i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 19 | \( 1 + 33.0iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (36.6 + 136. i)T + (-1.05e4 + 6.08e3i)T^{2} \) |
| 29 | \( 1 + (45.9 - 171. i)T + (-2.11e4 - 1.21e4i)T^{2} \) |
| 31 | \( 1 + (-69.6 + 18.6i)T + (2.57e4 - 1.48e4i)T^{2} \) |
| 37 | \( 1 + (203. + 203. i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 + (-46.9 - 175. i)T + (-5.96e4 + 3.44e4i)T^{2} \) |
| 43 | \( 1 + (245. - 141. i)T + (3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-92.2 - 159. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 - 80.6iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (207. + 119. i)T + (1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-131. - 35.3i)T + (1.96e5 + 1.13e5i)T^{2} \) |
| 67 | \( 1 + (-511. + 885. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (403. + 403. i)T + 3.57e5iT^{2} \) |
| 73 | \( 1 + (-264. - 264. i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 + (-814. - 218. i)T + (4.26e5 + 2.46e5i)T^{2} \) |
| 83 | \( 1 + (-671. + 387. i)T + (2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + 578.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (29.6 - 110. i)T + (-7.90e5 - 4.56e5i)T^{2} \) |
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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
−12.27687371878565006982238493123, −11.22997443674939422314458253417, −10.67491611406986936637506911843, −8.419732723020234770168818225142, −7.65913368067937019245027535535, −6.44345756664659014189761764702, −5.05986720700480136885912295415, −3.46609676190147165522735295681, −2.88585835046322850328753242798, −0.43155162233603736189543034865,
3.61082201825263528464577599010, 3.99679834452261784329072302453, 5.14169825075461676539162197907, 6.35996926999653330464963678710, 7.72390943472436885930339852042, 8.731434143973764656873711151419, 10.05084142237490593091739295486, 11.47792190760808239532693301685, 12.06863950738323943951477193547, 13.21774467292564400212292443070
