| L(s) = 1 | + 2i·5-s + (2.23 − 1.41i)7-s + 1.41·11-s − 4.47·17-s + 6.32·19-s + 3.16i·23-s + 25-s + 3.16·29-s + 5.65i·31-s + (2.82 + 4.47i)35-s − 4.47i·37-s − 4.47·41-s + 4i·43-s − 8·47-s + (3.00 − 6.32i)49-s + ⋯ |
| L(s) = 1 | + 0.894i·5-s + (0.845 − 0.534i)7-s + 0.426·11-s − 1.08·17-s + 1.45·19-s + 0.659i·23-s + 0.200·25-s + 0.587·29-s + 1.01i·31-s + (0.478 + 0.755i)35-s − 0.735i·37-s − 0.698·41-s + 0.609i·43-s − 1.16·47-s + (0.428 − 0.903i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.742 - 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.742 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.212550455\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.212550455\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.23 + 1.41i)T \) |
| good | 5 | \( 1 - 2iT - 5T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 - 6.32T + 19T^{2} \) |
| 23 | \( 1 - 3.16iT - 23T^{2} \) |
| 29 | \( 1 - 3.16T + 29T^{2} \) |
| 31 | \( 1 - 5.65iT - 31T^{2} \) |
| 37 | \( 1 + 4.47iT - 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 9.48T + 53T^{2} \) |
| 59 | \( 1 + 8.94iT - 59T^{2} \) |
| 61 | \( 1 - 8.48T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 - 9.48iT - 71T^{2} \) |
| 73 | \( 1 - 6.32iT - 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 + 6.32iT - 97T^{2} \) |
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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
−8.448350359079877300053711601161, −7.73159465798914559537657333773, −6.92836888255735900084624286407, −6.65718118811978508567859195133, −5.44436272927258463489908179171, −4.82223712953535444131568057881, −3.85028379205960830063738022081, −3.13335864231904509362101276878, −2.08056496632941336419480641476, −1.03579387247350341215705336896,
0.77627007633338196211317028281, 1.76358509724065347547878651284, 2.72200236696222066444856604833, 3.90096067829477677520621786955, 4.78342769544404680900200632190, 5.14676031469174319460071155825, 6.08994499315222454447131323835, 6.89632818548934147309736724059, 7.77862801316391907942751889876, 8.493547611520934195632748674663
