| L(s) = 1 | + (−1.44 + 1.60i)2-s + (0.0614 + 0.584i)3-s + (−0.278 − 2.65i)4-s + (−2.12 + 0.704i)5-s + (−1.02 − 0.746i)6-s + (−2.24 − 1.39i)7-s + (1.16 + 0.843i)8-s + (2.59 − 0.551i)9-s + (1.93 − 4.42i)10-s + (−5.51 − 1.17i)11-s + (1.53 − 0.325i)12-s + (−0.883 + 2.71i)13-s + (5.49 − 1.58i)14-s + (−0.542 − 1.19i)15-s + (2.17 − 0.463i)16-s + (−2.32 + 1.03i)17-s + ⋯ |
| L(s) = 1 | + (−1.02 + 1.13i)2-s + (0.0354 + 0.337i)3-s + (−0.139 − 1.32i)4-s + (−0.949 + 0.314i)5-s + (−0.419 − 0.304i)6-s + (−0.848 − 0.529i)7-s + (0.410 + 0.298i)8-s + (0.865 − 0.183i)9-s + (0.612 − 1.39i)10-s + (−1.66 − 0.353i)11-s + (0.442 − 0.0940i)12-s + (−0.244 + 0.753i)13-s + (1.46 − 0.422i)14-s + (−0.140 − 0.309i)15-s + (0.544 − 0.115i)16-s + (−0.563 + 0.250i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0292999 - 0.0370119i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0292999 - 0.0370119i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.12 - 0.704i)T \) |
| 7 | \( 1 + (2.24 + 1.39i)T \) |
| good | 2 | \( 1 + (1.44 - 1.60i)T + (-0.209 - 1.98i)T^{2} \) |
| 3 | \( 1 + (-0.0614 - 0.584i)T + (-2.93 + 0.623i)T^{2} \) |
| 11 | \( 1 + (5.51 + 1.17i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (0.883 - 2.71i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (2.32 - 1.03i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (-0.474 + 4.51i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (0.136 - 0.151i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (1.83 - 1.33i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (5.85 - 2.60i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (8.04 - 1.70i)T + (33.8 - 15.0i)T^{2} \) |
| 41 | \( 1 + (3.35 - 10.3i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 6.23T + 43T^{2} \) |
| 47 | \( 1 + (-5.73 - 2.55i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (1.34 + 12.8i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (1.58 + 1.75i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (4.98 - 5.54i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-3.43 + 1.53i)T + (44.8 - 49.7i)T^{2} \) |
| 71 | \( 1 + (-3.78 + 2.75i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.96 - 1.05i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (-4.60 - 2.04i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-4.00 - 2.90i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-2.37 + 2.63i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (-3.78 + 2.75i)T + (29.9 - 92.2i)T^{2} \) |
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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
−13.47273447502153701198781268648, −12.55007473234093456340045174108, −10.98760901511961706862706492993, −10.21327122462977358886218075751, −9.269707006783013220536943227214, −8.194654450034165925126788140193, −7.18353150018032069005582061557, −6.69297865768426484573411011791, −4.93348232278323696368904153165, −3.40309009972596445108278073125,
0.05672385989357376793932355667, 2.22976351802134509008649844712, 3.56998014320710825662629290634, 5.36925265430882974134904868217, 7.32916263507276113385174933745, 8.034008009459309347624716437596, 9.108344227643168945904185345284, 10.19826970513456456955278635966, 10.74956500978588069100610814437, 12.20628259189974857213303313055
