| L(s) = 1 | + 3.31i·3-s + 2·4-s + (1.5 + 1.65i)5-s − 8·9-s + 6.63i·12-s + (−5.5 + 4.97i)15-s + 4·16-s + (3 + 3.31i)20-s − 3.31i·23-s + (−0.5 + 4.97i)25-s − 16.5i·27-s + 5·31-s − 16·36-s − 9.94i·37-s + (−12 − 13.2i)45-s + ⋯ |
| L(s) = 1 | + 1.91i·3-s + 4-s + (0.670 + 0.741i)5-s − 2.66·9-s + 1.91i·12-s + (−1.42 + 1.28i)15-s + 16-s + (0.670 + 0.741i)20-s − 0.691i·23-s + (−0.100 + 0.994i)25-s − 3.19i·27-s + 0.898·31-s − 2.66·36-s − 1.63i·37-s + (−1.78 − 1.97i)45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.670 - 0.741i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.670 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.785692 + 1.77011i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.785692 + 1.77011i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.5 - 1.65i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2T^{2} \) |
| 3 | \( 1 - 3.31iT - 3T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 3.31iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 + 9.94iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 6.63iT - 47T^{2} \) |
| 53 | \( 1 - 13.2iT - 53T^{2} \) |
| 59 | \( 1 + 15T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 9.94iT - 67T^{2} \) |
| 71 | \( 1 + 3T + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 9T + 89T^{2} \) |
| 97 | \( 1 + 9.94iT - 97T^{2} \) |
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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
−10.67802042359854566449215005045, −10.33985578465369453467663329351, −9.502301252710149187502735559205, −8.635997787248410027129593456426, −7.35585356436051050074478985162, −6.16849001540665366062033611885, −5.57963650838486912801322515912, −4.34742651288809560520636681841, −3.23506279303977129601456255998, −2.42410438241969137702583412487,
1.12835459623987926917148295459, 1.98750238801095507182588716509, 3.00724456626015192435405198021, 5.17754805365538913671098916388, 6.16869289395445311960212154345, 6.60521237518476153560234420747, 7.67500229435343576420001945342, 8.215912503114003664649786779515, 9.284044306472577975284911193307, 10.50365325630705075796403638387
