L(s) = 1 | + 2.37i·2-s + 1.95i·3-s − 3.66·4-s + (0.479 − 2.18i)5-s − 4.66·6-s + 2.28i·7-s − 3.95i·8-s − 0.840·9-s + (5.19 + 1.14i)10-s + 1.12·11-s − 7.17i·12-s − 5.95i·13-s − 5.43·14-s + (4.27 + 0.940i)15-s + 2.09·16-s + 5.80i·17-s + ⋯ |
L(s) = 1 | + 1.68i·2-s + 1.13i·3-s − 1.83·4-s + (0.214 − 0.976i)5-s − 1.90·6-s + 0.863i·7-s − 1.39i·8-s − 0.280·9-s + (1.64 + 0.361i)10-s + 0.338·11-s − 2.07i·12-s − 1.65i·13-s − 1.45·14-s + (1.10 + 0.242i)15-s + 0.523·16-s + 1.40i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.214i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 - 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.108893 + 1.00305i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.108893 + 1.00305i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.479 + 2.18i)T \) |
| 23 | \( 1 - iT \) |
good | 2 | \( 1 - 2.37iT - 2T^{2} \) |
| 3 | \( 1 - 1.95iT - 3T^{2} \) |
| 7 | \( 1 - 2.28iT - 7T^{2} \) |
| 11 | \( 1 - 1.12T + 11T^{2} \) |
| 13 | \( 1 + 5.95iT - 13T^{2} \) |
| 17 | \( 1 - 5.80iT - 17T^{2} \) |
| 19 | \( 1 - 4.08T + 19T^{2} \) |
| 29 | \( 1 - 0.408T + 29T^{2} \) |
| 31 | \( 1 + 3.19T + 31T^{2} \) |
| 37 | \( 1 + 9.80iT - 37T^{2} \) |
| 41 | \( 1 - 6.27T + 41T^{2} \) |
| 43 | \( 1 + 7.75iT - 43T^{2} \) |
| 47 | \( 1 + 6.40iT - 47T^{2} \) |
| 53 | \( 1 - 6.73iT - 53T^{2} \) |
| 59 | \( 1 - 4.75T + 59T^{2} \) |
| 61 | \( 1 + 6.33T + 61T^{2} \) |
| 67 | \( 1 - 0.283iT - 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 + 9.61iT - 73T^{2} \) |
| 79 | \( 1 + 4.48T + 79T^{2} \) |
| 83 | \( 1 - 10.8iT - 83T^{2} \) |
| 89 | \( 1 + 5.68T + 89T^{2} \) |
| 97 | \( 1 - 11.0iT - 97T^{2} \) |
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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
−14.58474084123959755208240556738, −13.25664624055274704126806561614, −12.34488318098791564667308873126, −10.51656850836679121150672956589, −9.336111704148776990172713578344, −8.700506251679590464744058463536, −7.63989835367976680665340664785, −5.74132543623244504221758195139, −5.38005674754705759608137432761, −3.99893594056389816868909038101,
1.42939676877140458490427123424, 2.83555660150489837730819040837, 4.32929558696145820560478787902, 6.64728380596904064174942250890, 7.42847407166445297181581665105, 9.299929702622690237451935480432, 10.11003503590055307712284487832, 11.44687200453004858254690928588, 11.68780640141083658914923861794, 13.01594372965946494867125097818