| L(s) = 1 | + i·2-s − 4-s + (1.23 − 1.86i)5-s + i·7-s − i·8-s + (1.86 + 1.23i)10-s − 1.73·11-s + 5.46i·13-s − 14-s + 16-s + 4i·17-s − 5.73·19-s + (−1.23 + 1.86i)20-s − 1.73i·22-s − 2.46i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (0.550 − 0.834i)5-s + 0.377i·7-s − 0.353i·8-s + (0.590 + 0.389i)10-s − 0.522·11-s + 1.51i·13-s − 0.267·14-s + 0.250·16-s + 0.970i·17-s − 1.31·19-s + (−0.275 + 0.417i)20-s − 0.369i·22-s − 0.513i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.834 - 0.550i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.834 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.072018363\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.072018363\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + (-1.23 + 1.86i)T \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 + 1.73T + 11T^{2} \) |
| 13 | \( 1 - 5.46iT - 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 + 5.73T + 19T^{2} \) |
| 23 | \( 1 + 2.46iT - 23T^{2} \) |
| 29 | \( 1 - 1.46T + 29T^{2} \) |
| 31 | \( 1 + 0.267T + 31T^{2} \) |
| 37 | \( 1 - 3.19iT - 37T^{2} \) |
| 41 | \( 1 - 3.92T + 41T^{2} \) |
| 43 | \( 1 + 4.53iT - 43T^{2} \) |
| 47 | \( 1 - 11.4iT - 47T^{2} \) |
| 53 | \( 1 - 13.8iT - 53T^{2} \) |
| 59 | \( 1 - 4.92T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 - 12.3iT - 67T^{2} \) |
| 71 | \( 1 + 4.26T + 71T^{2} \) |
| 73 | \( 1 - 16.9iT - 73T^{2} \) |
| 79 | \( 1 + 8.92T + 79T^{2} \) |
| 83 | \( 1 + 9.46iT - 83T^{2} \) |
| 89 | \( 1 - T + 89T^{2} \) |
| 97 | \( 1 - 6.92iT - 97T^{2} \) |
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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
−9.231029763405207474599507697442, −8.732562029842110530982784437123, −8.158933003685856297454046316555, −7.09364026315952726012881606118, −6.18728016792368684901810669524, −5.78034176033413174026380146712, −4.54477589035331708067539913157, −4.25977023146467938515946561023, −2.53774314239683629089996338363, −1.48142931770240521163246854175,
0.38087903848068648536230412476, 1.97549496155003261729230608173, 2.87077661223126263852750066661, 3.57855830945405987133930477712, 4.81708775967335016100297767316, 5.59019180853021366207730855078, 6.47583881774773765618430052237, 7.44179136749050682927233180373, 8.093031287025754512664425634854, 9.103288962497194475459935560428
