| L(s) = 1 | + 3.26·3-s + (4.26 − 2.60i)5-s + 6.30i·7-s + 1.67·9-s + 4.53·11-s + 11.5·13-s + (13.9 − 8.52i)15-s − 1.08i·17-s + (16.8 + 8.76i)19-s + 20.5i·21-s + 31.5i·23-s + (11.3 − 22.2i)25-s − 23.9·27-s − 38.1i·29-s − 58.2i·31-s + ⋯ |
| L(s) = 1 | + 1.08·3-s + (0.853 − 0.521i)5-s + 0.900i·7-s + 0.186·9-s + 0.411·11-s + 0.887·13-s + (0.929 − 0.568i)15-s − 0.0638i·17-s + (0.887 + 0.461i)19-s + 0.980i·21-s + 1.37i·23-s + (0.455 − 0.890i)25-s − 0.886·27-s − 1.31i·29-s − 1.87i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0692i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0692i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.84306 + 0.0985231i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.84306 + 0.0985231i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-4.26 + 2.60i)T \) |
| 19 | \( 1 + (-16.8 - 8.76i)T \) |
| good | 3 | \( 1 - 3.26T + 9T^{2} \) |
| 7 | \( 1 - 6.30iT - 49T^{2} \) |
| 11 | \( 1 - 4.53T + 121T^{2} \) |
| 13 | \( 1 - 11.5T + 169T^{2} \) |
| 17 | \( 1 + 1.08iT - 289T^{2} \) |
| 23 | \( 1 - 31.5iT - 529T^{2} \) |
| 29 | \( 1 + 38.1iT - 841T^{2} \) |
| 31 | \( 1 + 58.2iT - 961T^{2} \) |
| 37 | \( 1 + 40.2T + 1.36e3T^{2} \) |
| 41 | \( 1 - 18.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 2.74iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 76.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 8.06T + 2.80e3T^{2} \) |
| 59 | \( 1 + 95.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 4.28T + 3.72e3T^{2} \) |
| 67 | \( 1 + 70.7T + 4.48e3T^{2} \) |
| 71 | \( 1 - 93.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 14.2iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.11iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 89.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 92.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 64.6T + 9.40e3T^{2} \) |
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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
−11.27180347069753805657145660001, −9.675894146374505258662666527852, −9.400675048870279107302874245882, −8.493197674531223538374850050470, −7.73636546403978613803305896463, −6.12688209345685643451305770141, −5.49775638519507848039610472942, −3.91777491629566328626767240461, −2.70890878417706198517302896344, −1.58138793390594229243230355461,
1.45350913640438680696178142705, 2.89751063531027252024522482782, 3.74081976373252399251693675293, 5.26514054408062902903227537663, 6.61612739116866538266989611984, 7.27463854073051010939486940933, 8.641152373033285793503204392976, 9.059616079759330855377190162583, 10.34606845434048134006403173101, 10.72959238420913314190000613521
