| L(s) = 1 | + (−1.26 + 0.642i)2-s + (0.742 − 0.117i)3-s + (1.17 − 1.61i)4-s + (−0.860 + 0.624i)6-s + (−2.71 − 2.71i)7-s + (−0.442 + 2.79i)8-s + (−8.02 + 2.60i)9-s + (−4.47 + 13.7i)11-s + (0.682 − 1.33i)12-s + (16.6 + 8.47i)13-s + (5.16 + 1.67i)14-s + (−1.23 − 3.80i)16-s + (−10.5 − 1.67i)17-s + (8.43 − 8.43i)18-s + (11.3 + 15.6i)19-s + ⋯ |
| L(s) = 1 | + (−0.630 + 0.321i)2-s + (0.247 − 0.0392i)3-s + (0.293 − 0.404i)4-s + (−0.143 + 0.104i)6-s + (−0.387 − 0.387i)7-s + (−0.0553 + 0.349i)8-s + (−0.891 + 0.289i)9-s + (−0.407 + 1.25i)11-s + (0.0568 − 0.111i)12-s + (1.27 + 0.651i)13-s + (0.368 + 0.119i)14-s + (−0.0772 − 0.237i)16-s + (−0.621 − 0.0985i)17-s + (0.468 − 0.468i)18-s + (0.597 + 0.822i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.402 - 0.915i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.402 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.455969 + 0.698388i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.455969 + 0.698388i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.26 - 0.642i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (-0.742 + 0.117i)T + (8.55 - 2.78i)T^{2} \) |
| 7 | \( 1 + (2.71 + 2.71i)T + 49iT^{2} \) |
| 11 | \( 1 + (4.47 - 13.7i)T + (-97.8 - 71.1i)T^{2} \) |
| 13 | \( 1 + (-16.6 - 8.47i)T + (99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (10.5 + 1.67i)T + (274. + 89.3i)T^{2} \) |
| 19 | \( 1 + (-11.3 - 15.6i)T + (-111. + 343. i)T^{2} \) |
| 23 | \( 1 + (-8.15 - 16.0i)T + (-310. + 427. i)T^{2} \) |
| 29 | \( 1 + (15.1 - 20.8i)T + (-259. - 799. i)T^{2} \) |
| 31 | \( 1 + (-6.46 + 4.70i)T + (296. - 913. i)T^{2} \) |
| 37 | \( 1 + (31.9 - 62.7i)T + (-804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-1.20 - 3.72i)T + (-1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 + (9.83 - 9.83i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (5.34 + 33.7i)T + (-2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (-3.64 + 0.577i)T + (2.67e3 - 868. i)T^{2} \) |
| 59 | \( 1 + (111. - 36.1i)T + (2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-9.28 + 28.5i)T + (-3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + (-62.2 - 9.85i)T + (4.26e3 + 1.38e3i)T^{2} \) |
| 71 | \( 1 + (-94.8 - 68.9i)T + (1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (35.3 + 69.4i)T + (-3.13e3 + 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-85.0 + 117. i)T + (-1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-1.65 + 10.4i)T + (-6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + (-3.88 - 1.26i)T + (6.40e3 + 4.65e3i)T^{2} \) |
| 97 | \( 1 + (16.6 + 105. i)T + (-8.94e3 + 2.90e3i)T^{2} \) |
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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
−11.93972058085562461142337733301, −11.04552657205325432875294310862, −10.08518088415688717973670878861, −9.172102576099191377251172237167, −8.274664504407792921061905824560, −7.27655024663565918011418567029, −6.31196329124256856752945291934, −5.03354366181424310657853184109, −3.42604026421707778190516487465, −1.74904875603360100437063610497,
0.51189682760530953772837255311, 2.66719782878877186535766717752, 3.56741558121047727330725379982, 5.57029108200721598764706564204, 6.44551002351801933838014229587, 7.975463757812684799331738236034, 8.732135802616082900380024199440, 9.341426083466341232433823305889, 10.91442088561946178927986193716, 11.09009746129102470024651643625
