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
−8.586304348317993976680235061451, −7.911440093447018201728767481188, −7.45007730426294186197555495315, −6.49391308558873249908605879730, −5.44563490409981224919899395881, −4.87801897141218522668038894698, −4.30149782414628818942195654886, −2.95603786749237570275872515913, −1.59326306690158223010000782155, −1.26914444411743333579933619512,
0.66704344503516258245990695946, 2.41128882848740614339774504738, 3.09981452886562667854046388613, 3.80010891634635878526074284637, 4.87491733704579402551952139191, 5.70169169198287296298946298126, 6.42380480854003759770549266281, 7.15925961588106868404236596845, 8.003781461702239975723125034716, 8.738218038289313996247204331137