L(s) = 1 | − 3.91i·5-s + (−2.03 − 1.68i)7-s + 1.62·11-s − 2.54·13-s + 2.94·17-s + 2.72·19-s − 8.57i·23-s − 10.3·25-s − 8.21·29-s − 8.08i·31-s + (−6.61 + 7.98i)35-s + 8.05i·37-s + 9.17·41-s − 9.05i·43-s − 1.17·47-s + ⋯ |
L(s) = 1 | − 1.75i·5-s + (−0.769 − 0.638i)7-s + 0.490·11-s − 0.706·13-s + 0.713·17-s + 0.624·19-s − 1.78i·23-s − 2.07·25-s − 1.52·29-s − 1.45i·31-s + (−1.11 + 1.34i)35-s + 1.32i·37-s + 1.43·41-s − 1.38i·43-s − 0.171·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.331i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 - 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.004282672\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.004282672\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.03 + 1.68i)T \) |
good | 5 | \( 1 + 3.91iT - 5T^{2} \) |
| 11 | \( 1 - 1.62T + 11T^{2} \) |
| 13 | \( 1 + 2.54T + 13T^{2} \) |
| 17 | \( 1 - 2.94T + 17T^{2} \) |
| 19 | \( 1 - 2.72T + 19T^{2} \) |
| 23 | \( 1 + 8.57iT - 23T^{2} \) |
| 29 | \( 1 + 8.21T + 29T^{2} \) |
| 31 | \( 1 + 8.08iT - 31T^{2} \) |
| 37 | \( 1 - 8.05iT - 37T^{2} \) |
| 41 | \( 1 - 9.17T + 41T^{2} \) |
| 43 | \( 1 + 9.05iT - 43T^{2} \) |
| 47 | \( 1 + 1.17T + 47T^{2} \) |
| 53 | \( 1 - 2.44T + 53T^{2} \) |
| 59 | \( 1 + 1.45iT - 59T^{2} \) |
| 61 | \( 1 + 9.74T + 61T^{2} \) |
| 67 | \( 1 + 7.35iT - 67T^{2} \) |
| 71 | \( 1 - 7.71iT - 71T^{2} \) |
| 73 | \( 1 + 15.0iT - 73T^{2} \) |
| 79 | \( 1 - 0.0913T + 79T^{2} \) |
| 83 | \( 1 - 2.03iT - 83T^{2} \) |
| 89 | \( 1 - 1.48T + 89T^{2} \) |
| 97 | \( 1 - 7.53iT - 97T^{2} \) |
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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
−7.939846227999579739193803762830, −7.53449078594216360239608966723, −6.48434476234163659165977924228, −5.75719526884661713983843816303, −4.94621877276624045215155871934, −4.28271990018536220226120758363, −3.58445416791285223047819756985, −2.31647624615318686525435299623, −1.10519036325180299190500802771, −0.31252006454490443703604891895,
1.68294177555647134886193106651, 2.79889273816273916546159310954, 3.24140124939405716296705418426, 4.00660504765562250419706504593, 5.51256940426624322256081291635, 5.82402972504373176585165144739, 6.78162550079918058720083856612, 7.31917219117607235020434273623, 7.78273215462079314250178210364, 9.138643898362460287194694729971