| L(s) = 1 | + (−5.43 − 1.76i)2-s + (−17.9 + 24.7i)3-s + (0.542 + 0.394i)4-s + (37.4 − 41.5i)5-s + (141. − 102. i)6-s − 28.6i·7-s + (105. + 144. i)8-s + (−214. − 660. i)9-s + (−276. + 159. i)10-s + (0.178 − 0.550i)11-s + (−19.5 + 6.34i)12-s + (444. − 144. i)13-s + (−50.5 + 155. i)14-s + (355. + 1.67e3i)15-s + (−322. − 993. i)16-s + (−591. − 814. i)17-s + ⋯ |
| L(s) = 1 | + (−0.960 − 0.312i)2-s + (−1.15 + 1.58i)3-s + (0.0169 + 0.0123i)4-s + (0.669 − 0.742i)5-s + (1.60 − 1.16i)6-s − 0.220i·7-s + (0.581 + 0.800i)8-s + (−0.883 − 2.71i)9-s + (−0.875 + 0.504i)10-s + (0.000446 − 0.00137i)11-s + (−0.0391 + 0.0127i)12-s + (0.729 − 0.236i)13-s + (−0.0689 + 0.212i)14-s + (0.407 + 1.92i)15-s + (−0.315 − 0.970i)16-s + (−0.496 − 0.683i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.348 + 0.937i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.348 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.367800 - 0.255765i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.367800 - 0.255765i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-37.4 + 41.5i)T \) |
| good | 2 | \( 1 + (5.43 + 1.76i)T + (25.8 + 18.8i)T^{2} \) |
| 3 | \( 1 + (17.9 - 24.7i)T + (-75.0 - 231. i)T^{2} \) |
| 7 | \( 1 + 28.6iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (-0.178 + 0.550i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (-444. + 144. i)T + (3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (591. + 814. i)T + (-4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-1.06e3 + 770. i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (3.09e3 + 1.00e3i)T + (5.20e6 + 3.78e6i)T^{2} \) |
| 29 | \( 1 + (1.53e3 + 1.11e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-3.73e3 + 2.71e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (829. - 269. i)T + (5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (3.98e3 + 1.22e4i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 + 2.72e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (4.78e3 - 6.58e3i)T + (-7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (8.34e3 - 1.14e4i)T + (-1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-9.82e3 - 3.02e4i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-8.92e3 + 2.74e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + (2.37e4 + 3.26e4i)T + (-4.17e8 + 1.28e9i)T^{2} \) |
| 71 | \( 1 + (2.48e4 + 1.80e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (1.48e4 + 4.83e3i)T + (1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-1.38e4 - 1.00e4i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (6.55e4 + 9.01e4i)T + (-1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + (1.92e4 - 5.91e4i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (2.67e4 - 3.67e4i)T + (-2.65e9 - 8.16e9i)T^{2} \) |
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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
−16.56879958464593043560011013478, −15.66121967986910787663358051680, −13.85757205963971612057629256254, −11.79746110349001781688374007645, −10.63232118535150818340813589757, −9.748426194695725246879812229791, −8.848496559240694204921627541212, −5.78066512597706186040682698662, −4.53733032013823218016624825752, −0.51444210936831395441594675073,
1.54935544554383232488607857066, 5.94515437337952557578426272494, 6.97458473692256792545497870054, 8.246298000641168122418937449201, 10.24143152223596069664430395257, 11.52949257899231985316007106722, 12.97434317813495322947944920265, 13.88014735030538666706866057889, 16.19513000976553450437149688902, 17.30177752593100063009763473580
