L(s) = 1 | + (9.28 − 9.28i)3-s + (42.6 + 36.0i)5-s + (−105. − 105. i)7-s + 70.7i·9-s + 344. i·11-s + (707. + 707. i)13-s + (731. − 61.3i)15-s + (1.26e3 − 1.26e3i)17-s + 438.·19-s − 1.96e3·21-s + (−1.72e3 + 1.72e3i)23-s + (520. + 3.08e3i)25-s + (2.91e3 + 2.91e3i)27-s + 2.51e3i·29-s − 7.14e3i·31-s + ⋯ |
L(s) = 1 | + (0.595 − 0.595i)3-s + (0.763 + 0.645i)5-s + (−0.816 − 0.816i)7-s + 0.291i·9-s + 0.859i·11-s + (1.16 + 1.16i)13-s + (0.839 − 0.0703i)15-s + (1.05 − 1.05i)17-s + 0.278·19-s − 0.971·21-s + (−0.679 + 0.679i)23-s + (0.166 + 0.986i)25-s + (0.768 + 0.768i)27-s + 0.554i·29-s − 1.33i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.310i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.950 - 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.658406422\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.658406422\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-42.6 - 36.0i)T \) |
good | 3 | \( 1 + (-9.28 + 9.28i)T - 243iT^{2} \) |
| 7 | \( 1 + (105. + 105. i)T + 1.68e4iT^{2} \) |
| 11 | \( 1 - 344. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (-707. - 707. i)T + 3.71e5iT^{2} \) |
| 17 | \( 1 + (-1.26e3 + 1.26e3i)T - 1.41e6iT^{2} \) |
| 19 | \( 1 - 438.T + 2.47e6T^{2} \) |
| 23 | \( 1 + (1.72e3 - 1.72e3i)T - 6.43e6iT^{2} \) |
| 29 | \( 1 - 2.51e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 7.14e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + (3.36e3 - 3.36e3i)T - 6.93e7iT^{2} \) |
| 41 | \( 1 - 6.96e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + (-1.63e4 + 1.63e4i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + (-1.30e4 - 1.30e4i)T + 2.29e8iT^{2} \) |
| 53 | \( 1 + (-2.00e4 - 2.00e4i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 - 8.41e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.29e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + (-395. - 395. i)T + 1.35e9iT^{2} \) |
| 71 | \( 1 - 4.08e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (4.56e4 + 4.56e4i)T + 2.07e9iT^{2} \) |
| 79 | \( 1 + 5.84e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-2.69e4 + 2.69e4i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 - 6.19e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (1.10e4 - 1.10e4i)T - 8.58e9iT^{2} \) |
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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
−12.17409502133140527562389585838, −10.89962891457869383956839453365, −9.906965030754257507510853574971, −9.135623774514804789149588780481, −7.50258122259014586796350153091, −7.00255325377758710180487249546, −5.76208807610790333108515671474, −3.93428224213426673757587951389, −2.60870918179524455051715736523, −1.35857038319010044384302043991,
0.933270648023861040921065627100, 2.82907750635253006250112616136, 3.80499197063092105998737898436, 5.66809759071648031572390275729, 6.12336223343240814750730273416, 8.307664025955770678916762951489, 8.804720643531841837438126913344, 9.822805382481269672487503747421, 10.58458814689793181089726079933, 12.23899417655979869564563737581