| L(s) = 1 | + (−0.0924 + 0.732i)2-s + (−1.70 + 1.41i)3-s + (1.40 + 0.361i)4-s + (2.23 − 0.154i)5-s + (−0.875 − 1.37i)6-s + (−2.87 + 0.934i)7-s + (−0.938 + 2.37i)8-s + (0.357 − 1.87i)9-s + (−0.0931 + 1.64i)10-s + (−1.76 − 0.222i)11-s + (−2.91 + 1.37i)12-s + (3.74 + 0.714i)13-s + (−0.417 − 2.19i)14-s + (−3.58 + 3.41i)15-s + (0.902 + 0.496i)16-s + (−0.634 − 2.46i)17-s + ⋯ |
| L(s) = 1 | + (−0.0653 + 0.517i)2-s + (−0.985 + 0.815i)3-s + (0.704 + 0.180i)4-s + (0.997 − 0.0691i)5-s + (−0.357 − 0.563i)6-s + (−1.08 + 0.353i)7-s + (−0.331 + 0.838i)8-s + (0.119 − 0.624i)9-s + (−0.0294 + 0.520i)10-s + (−0.531 − 0.0671i)11-s + (−0.842 + 0.396i)12-s + (1.03 + 0.198i)13-s + (−0.111 − 0.585i)14-s + (−0.926 + 0.881i)15-s + (0.225 + 0.124i)16-s + (−0.153 − 0.598i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.582911 + 0.746723i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.582911 + 0.746723i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.23 + 0.154i)T \) |
| good | 2 | \( 1 + (0.0924 - 0.732i)T + (-1.93 - 0.497i)T^{2} \) |
| 3 | \( 1 + (1.70 - 1.41i)T + (0.562 - 2.94i)T^{2} \) |
| 7 | \( 1 + (2.87 - 0.934i)T + (5.66 - 4.11i)T^{2} \) |
| 11 | \( 1 + (1.76 + 0.222i)T + (10.6 + 2.73i)T^{2} \) |
| 13 | \( 1 + (-3.74 - 0.714i)T + (12.0 + 4.78i)T^{2} \) |
| 17 | \( 1 + (0.634 + 2.46i)T + (-14.8 + 8.18i)T^{2} \) |
| 19 | \( 1 + (1.15 - 1.39i)T + (-3.56 - 18.6i)T^{2} \) |
| 23 | \( 1 + (-6.08 + 6.48i)T + (-1.44 - 22.9i)T^{2} \) |
| 29 | \( 1 + (0.162 + 2.58i)T + (-28.7 + 3.63i)T^{2} \) |
| 31 | \( 1 + (-8.08 + 2.07i)T + (27.1 - 14.9i)T^{2} \) |
| 37 | \( 1 + (0.810 - 1.47i)T + (-19.8 - 31.2i)T^{2} \) |
| 41 | \( 1 + (5.33 - 5.01i)T + (2.57 - 40.9i)T^{2} \) |
| 43 | \( 1 + (-4.98 - 6.86i)T + (-13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (3.77 + 9.54i)T + (-34.2 + 32.1i)T^{2} \) |
| 53 | \( 1 + (-0.00179 - 0.00113i)T + (22.5 + 47.9i)T^{2} \) |
| 59 | \( 1 + (4.16 + 8.85i)T + (-37.6 + 45.4i)T^{2} \) |
| 61 | \( 1 + (4.96 + 4.66i)T + (3.83 + 60.8i)T^{2} \) |
| 67 | \( 1 + (-3.00 - 0.188i)T + (66.4 + 8.39i)T^{2} \) |
| 71 | \( 1 + (9.18 - 3.63i)T + (51.7 - 48.6i)T^{2} \) |
| 73 | \( 1 + (6.61 + 3.11i)T + (46.5 + 56.2i)T^{2} \) |
| 79 | \( 1 + (-3.35 - 4.05i)T + (-14.8 + 77.6i)T^{2} \) |
| 83 | \( 1 + (-7.00 - 5.79i)T + (15.5 + 81.5i)T^{2} \) |
| 89 | \( 1 + (-0.108 + 0.229i)T + (-56.7 - 68.5i)T^{2} \) |
| 97 | \( 1 + (18.5 - 1.16i)T + (96.2 - 12.1i)T^{2} \) |
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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
−13.63546049875772715068935141458, −12.64406821790903027894960568039, −11.43005379569155858533534232898, −10.58205672292393551693109885370, −9.701700122177864206439108431083, −8.450009234931829819692272609003, −6.53934184926463735193371851780, −6.12840833474845525492303274175, −4.95471318544906679194119538974, −2.81424460123349199252166810925,
1.31838077503550731512876342513, 3.10673323736730324536489475541, 5.64683200221147656923816957391, 6.39933350378258672624420619139, 7.14791058186017601694843914895, 9.174235020886337713606003493557, 10.40110114103473132488078825343, 10.91813054961197230184243406297, 12.12534147468973067005167634741, 13.03134337320416632180458133234
