L(s) = 1 | + i·3-s + 3.16i·5-s + 1.74·7-s − 9-s − 6.47i·11-s + 1.41i·13-s − 3.16·15-s + 2.47·17-s − 6.47i·19-s + 1.74i·21-s + 5.65·23-s − 5.00·25-s − i·27-s − 5.99i·29-s + 3.90·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 1.41i·5-s + 0.660·7-s − 0.333·9-s − 1.95i·11-s + 0.392i·13-s − 0.816·15-s + 0.599·17-s − 1.48i·19-s + 0.381i·21-s + 1.17·23-s − 1.00·25-s − 0.192i·27-s − 1.11i·29-s + 0.702·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.927618494\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.927618494\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
good | 5 | \( 1 - 3.16iT - 5T^{2} \) |
| 7 | \( 1 - 1.74T + 7T^{2} \) |
| 11 | \( 1 + 6.47iT - 11T^{2} \) |
| 13 | \( 1 - 1.41iT - 13T^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 19 | \( 1 + 6.47iT - 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 + 5.99iT - 29T^{2} \) |
| 31 | \( 1 - 3.90T + 31T^{2} \) |
| 37 | \( 1 + 10.5iT - 37T^{2} \) |
| 41 | \( 1 - 2.47T + 41T^{2} \) |
| 43 | \( 1 + 1.52iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 11.6iT - 53T^{2} \) |
| 59 | \( 1 + 8.94iT - 59T^{2} \) |
| 61 | \( 1 - 2.08iT - 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 9.15T + 71T^{2} \) |
| 73 | \( 1 + 2.94T + 73T^{2} \) |
| 79 | \( 1 - 7.40T + 79T^{2} \) |
| 83 | \( 1 + 6.47iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−8.738218038289313996247204331137, −8.003781461702239975723125034716, −7.15925961588106868404236596845, −6.42380480854003759770549266281, −5.70169169198287296298946298126, −4.87491733704579402551952139191, −3.80010891634635878526074284637, −3.09981452886562667854046388613, −2.41128882848740614339774504738, −0.66704344503516258245990695946,
1.26914444411743333579933619512, 1.59326306690158223010000782155, 2.95603786749237570275872515913, 4.30149782414628818942195654886, 4.87801897141218522668038894698, 5.44563490409981224919899395881, 6.49391308558873249908605879730, 7.45007730426294186197555495315, 7.911440093447018201728767481188, 8.586304348317993976680235061451