| L(s) = 1 | − i·2-s − 4-s − 2.68i·5-s + (−0.592 − 2.57i)7-s + i·8-s − 2.68·10-s + 1.22·11-s − 5.01·13-s + (−2.57 + 0.592i)14-s + 16-s + 5.15i·17-s + (−2.75 + 3.37i)19-s + 2.68i·20-s − 1.22i·22-s − 3.18·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 1.20i·5-s + (−0.224 − 0.974i)7-s + 0.353i·8-s − 0.850·10-s + 0.370·11-s − 1.39·13-s + (−0.689 + 0.158i)14-s + 0.250·16-s + 1.25i·17-s + (−0.632 + 0.774i)19-s + 0.601i·20-s − 0.261i·22-s − 0.664·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1610817931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1610817931\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.592 + 2.57i)T \) |
| 19 | \( 1 + (2.75 - 3.37i)T \) |
| good | 5 | \( 1 + 2.68iT - 5T^{2} \) |
| 11 | \( 1 - 1.22T + 11T^{2} \) |
| 13 | \( 1 + 5.01T + 13T^{2} \) |
| 17 | \( 1 - 5.15iT - 17T^{2} \) |
| 23 | \( 1 + 3.18T + 23T^{2} \) |
| 29 | \( 1 - 4.47iT - 29T^{2} \) |
| 31 | \( 1 + 6.61T + 31T^{2} \) |
| 37 | \( 1 - 3.74iT - 37T^{2} \) |
| 41 | \( 1 - 0.873T + 41T^{2} \) |
| 43 | \( 1 - 8.11T + 43T^{2} \) |
| 47 | \( 1 + 7.36iT - 47T^{2} \) |
| 53 | \( 1 - 1.58iT - 53T^{2} \) |
| 59 | \( 1 - 2.46T + 59T^{2} \) |
| 61 | \( 1 + 9.69iT - 61T^{2} \) |
| 67 | \( 1 - 0.728iT - 67T^{2} \) |
| 71 | \( 1 - 5.14iT - 71T^{2} \) |
| 73 | \( 1 + 6.11iT - 73T^{2} \) |
| 79 | \( 1 - 9.74iT - 79T^{2} \) |
| 83 | \( 1 - 1.81iT - 83T^{2} \) |
| 89 | \( 1 + 3.38T + 89T^{2} \) |
| 97 | \( 1 - 3.22T + 97T^{2} \) |
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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
−9.196128027374020893156171625654, −8.440534349718050949071353988575, −7.76709849488276139966238480902, −6.85817495573315053607819965417, −5.79719146444597833679180830026, −4.92599853320441972503343082044, −4.17912321201259402920552441545, −3.58665929475802660163214009529, −2.11273199845187998479557587924, −1.22654358852326577417090405808,
0.05572776636192827014778208392, 2.33286756827249506264521510775, 2.83144294848658802420043600236, 4.08046458411207742255898097436, 5.03548744994161658038496373632, 5.84752981681271569251804245464, 6.58435155969722564813930228573, 7.25415741284225078813297603455, 7.76959189152539931658101093606, 8.993643376487618671795425732314
