L(s) = 1 | + 4.72·7-s − 3.93i·11-s + 3.46i·13-s + 3.51·17-s + 5.44i·19-s + 7.11·23-s + 3.66i·29-s − 5.23·31-s + 0.414i·37-s − 3.00·41-s + 5.34i·43-s + 0.925·47-s + 15.3·49-s − 0.233i·53-s + 14.3i·59-s + ⋯ |
L(s) = 1 | + 1.78·7-s − 1.18i·11-s + 0.961i·13-s + 0.852·17-s + 1.24i·19-s + 1.48·23-s + 0.681i·29-s − 0.940·31-s + 0.0681i·37-s − 0.469·41-s + 0.815i·43-s + 0.135·47-s + 2.18·49-s − 0.0320i·53-s + 1.87i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 - 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.826 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.763514326\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.763514326\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 4.72T + 7T^{2} \) |
| 11 | \( 1 + 3.93iT - 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 3.51T + 17T^{2} \) |
| 19 | \( 1 - 5.44iT - 19T^{2} \) |
| 23 | \( 1 - 7.11T + 23T^{2} \) |
| 29 | \( 1 - 3.66iT - 29T^{2} \) |
| 31 | \( 1 + 5.23T + 31T^{2} \) |
| 37 | \( 1 - 0.414iT - 37T^{2} \) |
| 41 | \( 1 + 3.00T + 41T^{2} \) |
| 43 | \( 1 - 5.34iT - 43T^{2} \) |
| 47 | \( 1 - 0.925T + 47T^{2} \) |
| 53 | \( 1 + 0.233iT - 53T^{2} \) |
| 59 | \( 1 - 14.3iT - 59T^{2} \) |
| 61 | \( 1 + 0.118iT - 61T^{2} \) |
| 67 | \( 1 - 13.4iT - 67T^{2} \) |
| 71 | \( 1 - 2.19T + 71T^{2} \) |
| 73 | \( 1 - 0.563T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 + 11.3iT - 83T^{2} \) |
| 89 | \( 1 + 8.88T + 89T^{2} \) |
| 97 | \( 1 + 7.27T + 97T^{2} \) |
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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
−7.944399572236503767885195698662, −7.47431101510184657298637849291, −6.64602603961541153511644006125, −5.63529763448764693926389633774, −5.31099418527664057115999576566, −4.43474579165290600558039672627, −3.70899439638878887139888318845, −2.80787782661585174411369540610, −1.62416351400831302676995478407, −1.15096389018395017848759825043,
0.74739031576930957479118405651, 1.72248972396766007845436634087, 2.48005389261664018262695053504, 3.50044678053235752031985391657, 4.48249320097736013325881573777, 5.16932220359920368849545923167, 5.31897874348328413756524346150, 6.62819890862945486996859493266, 7.34017046387391200473015682063, 7.79115768999110016923829862956