L(s) = 1 | + (−0.505 − 1.32i)2-s + (0.658 − 0.658i)3-s + (−1.48 + 1.33i)4-s + (0.820 + 2.07i)5-s + (−1.20 − 0.536i)6-s + (1.89 + 1.89i)7-s + (2.51 + 1.28i)8-s + 2.13i·9-s + (2.33 − 2.13i)10-s + 1.63i·11-s + (−0.0997 + 1.85i)12-s + (−0.707 − 0.707i)13-s + (1.54 − 3.45i)14-s + (1.90 + 0.828i)15-s + (0.428 − 3.97i)16-s + (2.78 − 2.78i)17-s + ⋯ |
L(s) = 1 | + (−0.357 − 0.933i)2-s + (0.379 − 0.379i)3-s + (−0.743 + 0.668i)4-s + (0.367 + 0.930i)5-s + (−0.490 − 0.218i)6-s + (0.714 + 0.714i)7-s + (0.890 + 0.455i)8-s + 0.711i·9-s + (0.737 − 0.675i)10-s + 0.493i·11-s + (−0.0287 + 0.536i)12-s + (−0.196 − 0.196i)13-s + (0.411 − 0.923i)14-s + (0.492 + 0.213i)15-s + (0.107 − 0.994i)16-s + (0.674 − 0.674i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.21927 - 0.141640i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.21927 - 0.141640i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.505 + 1.32i)T \) |
| 5 | \( 1 + (-0.820 - 2.07i)T \) |
| 13 | \( 1 + (0.707 + 0.707i)T \) |
good | 3 | \( 1 + (-0.658 + 0.658i)T - 3iT^{2} \) |
| 7 | \( 1 + (-1.89 - 1.89i)T + 7iT^{2} \) |
| 11 | \( 1 - 1.63iT - 11T^{2} \) |
| 17 | \( 1 + (-2.78 + 2.78i)T - 17iT^{2} \) |
| 19 | \( 1 + 3.52T + 19T^{2} \) |
| 23 | \( 1 + (0.868 - 0.868i)T - 23iT^{2} \) |
| 29 | \( 1 + 7.97iT - 29T^{2} \) |
| 31 | \( 1 + 7.80iT - 31T^{2} \) |
| 37 | \( 1 + (-4.02 + 4.02i)T - 37iT^{2} \) |
| 41 | \( 1 - 5.49T + 41T^{2} \) |
| 43 | \( 1 + (5.16 - 5.16i)T - 43iT^{2} \) |
| 47 | \( 1 + (-8.12 - 8.12i)T + 47iT^{2} \) |
| 53 | \( 1 + (0.551 + 0.551i)T + 53iT^{2} \) |
| 59 | \( 1 - 8.09T + 59T^{2} \) |
| 61 | \( 1 + 8.43T + 61T^{2} \) |
| 67 | \( 1 + (1.32 + 1.32i)T + 67iT^{2} \) |
| 71 | \( 1 + 6.50iT - 71T^{2} \) |
| 73 | \( 1 + (4.95 + 4.95i)T + 73iT^{2} \) |
| 79 | \( 1 + 14.6T + 79T^{2} \) |
| 83 | \( 1 + (-6.57 + 6.57i)T - 83iT^{2} \) |
| 89 | \( 1 - 13.8iT - 89T^{2} \) |
| 97 | \( 1 + (0.0106 - 0.0106i)T - 97iT^{2} \) |
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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
−11.75214113872075906027130683972, −11.06835411303910594916636816684, −10.09979767057635329008829319587, −9.272006311432950000087331772245, −7.992491604562101840424616824964, −7.50588955675265871448442485524, −5.79009638202530968830504154878, −4.43329234980130386084007371012, −2.71307087072773745901559556812, −2.03270922744866953076103109400,
1.22168165305285123120545359242, 3.89488621498352667283099230041, 4.86836574933559323149736676361, 5.98089396343279907700084979677, 7.16415975633914989555714308639, 8.463425279023605228824900773815, 8.761139830246828263259741760892, 9.939810812735016060392941842701, 10.66307050526341765223948091797, 12.19685514223921232065092234326