| L(s) = 1 | + (−0.307 + 2.43i)2-s + (−1.96 + 1.62i)3-s + (−3.90 − 1.00i)4-s + (0.490 + 2.18i)5-s + (−3.35 − 5.29i)6-s + (3.88 − 1.26i)7-s + (1.84 − 4.65i)8-s + (0.657 − 3.44i)9-s + (−5.46 + 0.523i)10-s + (−2.63 − 0.332i)11-s + (9.31 − 4.38i)12-s + (−1.00 − 0.191i)13-s + (1.88 + 9.85i)14-s + (−4.51 − 3.49i)15-s + (3.69 + 2.03i)16-s + (1.70 + 6.63i)17-s + ⋯ |
| L(s) = 1 | + (−0.217 + 1.72i)2-s + (−1.13 + 0.939i)3-s + (−1.95 − 0.501i)4-s + (0.219 + 0.975i)5-s + (−1.37 − 2.16i)6-s + (1.46 − 0.477i)7-s + (0.651 − 1.64i)8-s + (0.219 − 1.14i)9-s + (−1.72 + 0.165i)10-s + (−0.794 − 0.100i)11-s + (2.69 − 1.26i)12-s + (−0.277 − 0.0529i)13-s + (0.502 + 2.63i)14-s + (−1.16 − 0.901i)15-s + (0.924 + 0.508i)16-s + (0.413 + 1.60i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.764 + 0.645i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.764 + 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.223091 - 0.610170i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.223091 - 0.610170i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.490 - 2.18i)T \) |
| good | 2 | \( 1 + (0.307 - 2.43i)T + (-1.93 - 0.497i)T^{2} \) |
| 3 | \( 1 + (1.96 - 1.62i)T + (0.562 - 2.94i)T^{2} \) |
| 7 | \( 1 + (-3.88 + 1.26i)T + (5.66 - 4.11i)T^{2} \) |
| 11 | \( 1 + (2.63 + 0.332i)T + (10.6 + 2.73i)T^{2} \) |
| 13 | \( 1 + (1.00 + 0.191i)T + (12.0 + 4.78i)T^{2} \) |
| 17 | \( 1 + (-1.70 - 6.63i)T + (-14.8 + 8.18i)T^{2} \) |
| 19 | \( 1 + (-1.39 + 1.68i)T + (-3.56 - 18.6i)T^{2} \) |
| 23 | \( 1 + (0.574 - 0.611i)T + (-1.44 - 22.9i)T^{2} \) |
| 29 | \( 1 + (0.0172 + 0.273i)T + (-28.7 + 3.63i)T^{2} \) |
| 31 | \( 1 + (-7.12 + 1.83i)T + (27.1 - 14.9i)T^{2} \) |
| 37 | \( 1 + (3.93 - 7.15i)T + (-19.8 - 31.2i)T^{2} \) |
| 41 | \( 1 + (-3.27 + 3.07i)T + (2.57 - 40.9i)T^{2} \) |
| 43 | \( 1 + (-0.821 - 1.13i)T + (-13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (-0.296 - 0.748i)T + (-34.2 + 32.1i)T^{2} \) |
| 53 | \( 1 + (-6.58 - 4.17i)T + (22.5 + 47.9i)T^{2} \) |
| 59 | \( 1 + (3.52 + 7.49i)T + (-37.6 + 45.4i)T^{2} \) |
| 61 | \( 1 + (-8.34 - 7.83i)T + (3.83 + 60.8i)T^{2} \) |
| 67 | \( 1 + (-2.68 - 0.169i)T + (66.4 + 8.39i)T^{2} \) |
| 71 | \( 1 + (2.00 - 0.795i)T + (51.7 - 48.6i)T^{2} \) |
| 73 | \( 1 + (12.0 + 5.66i)T + (46.5 + 56.2i)T^{2} \) |
| 79 | \( 1 + (-2.81 - 3.40i)T + (-14.8 + 77.6i)T^{2} \) |
| 83 | \( 1 + (2.89 + 2.39i)T + (15.5 + 81.5i)T^{2} \) |
| 89 | \( 1 + (-4.52 + 9.61i)T + (-56.7 - 68.5i)T^{2} \) |
| 97 | \( 1 + (-5.93 + 0.373i)T + (96.2 - 12.1i)T^{2} \) |
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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
−14.50637189304374770747745700081, −13.54325775721419417605539532291, −11.64211735667436203995450186532, −10.62787379782314504799077376838, −10.02814178196041539979057911697, −8.323387107619213639063286104071, −7.44596242101939627139787669546, −6.16842258282801079805283215083, −5.32281660358877259160021595404, −4.34509512871999989877194532308,
0.926766409757627921853845990413, 2.21642731161242566484112742699, 4.77215604215522019353407741325, 5.42846284781726257386503293911, 7.61026231361124739499204337594, 8.707894688495956559464159356319, 9.958581407968218522099515293013, 11.12884331257738886715487490534, 11.89116424486087614812006575253, 12.23713870943184620206584816095
