L(s) = 1 | + (0.0382 − 0.0661i)5-s + (2.16 + 1.51i)7-s + (−4.66 + 2.69i)11-s − 5.31i·13-s + (1.89 + 3.27i)17-s + (−4.33 − 2.50i)19-s + (2.02 + 1.16i)23-s + (2.49 + 4.32i)25-s + 10.2i·29-s + (−4.97 + 2.87i)31-s + (0.183 − 0.0852i)35-s + (0.354 − 0.613i)37-s + 6.59·41-s − 1.43·43-s + (1.46 − 2.53i)47-s + ⋯ |
L(s) = 1 | + (0.0170 − 0.0295i)5-s + (0.818 + 0.574i)7-s + (−1.40 + 0.811i)11-s − 1.47i·13-s + (0.458 + 0.794i)17-s + (−0.995 − 0.574i)19-s + (0.422 + 0.243i)23-s + (0.499 + 0.865i)25-s + 1.89i·29-s + (−0.893 + 0.516i)31-s + (0.0309 − 0.0144i)35-s + (0.0582 − 0.100i)37-s + 1.03·41-s − 0.218·43-s + (0.213 − 0.369i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.303 - 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.303 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.212417682\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.212417682\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.16 - 1.51i)T \) |
good | 5 | \( 1 + (-0.0382 + 0.0661i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (4.66 - 2.69i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 5.31iT - 13T^{2} \) |
| 17 | \( 1 + (-1.89 - 3.27i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.33 + 2.50i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.02 - 1.16i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 10.2iT - 29T^{2} \) |
| 31 | \( 1 + (4.97 - 2.87i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.354 + 0.613i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 6.59T + 41T^{2} \) |
| 43 | \( 1 + 1.43T + 43T^{2} \) |
| 47 | \( 1 + (-1.46 + 2.53i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (10.4 - 6.05i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.289 - 0.502i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.40 - 1.38i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.63 + 4.56i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 3.32iT - 71T^{2} \) |
| 73 | \( 1 + (6.17 - 3.56i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.469 - 0.812i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 + (-1.51 + 2.62i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 7.13iT - 97T^{2} \) |
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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
−9.058059948754848271619358567625, −8.489326670914831272893639462866, −7.68432309366955493484463736552, −7.20245473816742579253515298607, −5.88629994554222809370700190006, −5.23372005210899548207540127094, −4.73964590265112188985845242457, −3.34448547918381912711962227199, −2.51839998375635485339673024478, −1.42019594488899889620249561108,
0.40850208710211052638862502560, 1.89660729710654213747643275727, 2.80191722068047936711725469171, 4.10695163591380395285041634896, 4.65589687636887123796425540762, 5.63775322756763114034037362957, 6.43770535515147769782488164858, 7.40242027134712626714235058713, 8.013190746052146844019167285978, 8.636060123650590785858572234635