L(s) = 1 | − 0.0797·3-s + 0.579i·5-s − i·7-s − 2.99·9-s − i·11-s + (−1.19 + 3.40i)13-s − 0.0461i·15-s + 6.63·17-s − 3.33i·19-s + 0.0797i·21-s − 8.63·23-s + 4.66·25-s + 0.477·27-s + 0.664·29-s + 0.646i·31-s + ⋯ |
L(s) = 1 | − 0.0460·3-s + 0.258i·5-s − 0.377i·7-s − 0.997·9-s − 0.301i·11-s + (−0.331 + 0.943i)13-s − 0.0119i·15-s + 1.61·17-s − 0.764i·19-s + 0.0173i·21-s − 1.79·23-s + 0.932·25-s + 0.0919·27-s + 0.123·29-s + 0.116i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8735706217\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8735706217\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + iT \) |
| 13 | \( 1 + (1.19 - 3.40i)T \) |
good | 3 | \( 1 + 0.0797T + 3T^{2} \) |
| 5 | \( 1 - 0.579iT - 5T^{2} \) |
| 17 | \( 1 - 6.63T + 17T^{2} \) |
| 19 | \( 1 + 3.33iT - 19T^{2} \) |
| 23 | \( 1 + 8.63T + 23T^{2} \) |
| 29 | \( 1 - 0.664T + 29T^{2} \) |
| 31 | \( 1 - 0.646iT - 31T^{2} \) |
| 37 | \( 1 - 0.0692iT - 37T^{2} \) |
| 41 | \( 1 + 0.306iT - 41T^{2} \) |
| 43 | \( 1 + 5.52T + 43T^{2} \) |
| 47 | \( 1 - 2.26iT - 47T^{2} \) |
| 53 | \( 1 - 4.29T + 53T^{2} \) |
| 59 | \( 1 + 11.0iT - 59T^{2} \) |
| 61 | \( 1 + 2.98T + 61T^{2} \) |
| 67 | \( 1 + 2.33iT - 67T^{2} \) |
| 71 | \( 1 - 3.49iT - 71T^{2} \) |
| 73 | \( 1 + 9.99iT - 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 + 9.43iT - 83T^{2} \) |
| 89 | \( 1 + 6.81iT - 89T^{2} \) |
| 97 | \( 1 + 14.5iT - 97T^{2} \) |
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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
−8.206705470873189196656806927906, −7.53739018958279996658923428232, −6.72143889226714179099435578993, −6.04185839605392275898966065552, −5.29186728184344422530876301297, −4.43707531234459214659885487785, −3.47390759336104023512928489425, −2.78874424467150925592293133883, −1.65801452088506252664187787129, −0.26857893961171878438114092613,
1.13257263793074191872450730827, 2.38675806651399168795622734862, 3.17853360312447480362207780842, 4.02716246833900386273452973869, 5.22444710942714779927700398450, 5.58584971262915994876599475284, 6.27154221917607384167949063426, 7.37224276723410274226363933487, 8.117920005884189358472644048400, 8.419105322003407006376174898173