L(s) = 1 | + (−0.984 − 0.173i)2-s + (1.27 + 1.16i)3-s + (0.939 + 0.342i)4-s + (2.18 − 0.491i)5-s + (−1.05 − 1.37i)6-s + (−3.69 + 2.13i)7-s + (−0.866 − 0.5i)8-s + (0.268 + 2.98i)9-s + (−2.23 + 0.105i)10-s + (−2.36 − 1.36i)11-s + (0.801 + 1.53i)12-s + (−2.74 + 2.30i)13-s + (4.01 − 1.46i)14-s + (3.36 + 1.92i)15-s + (0.766 + 0.642i)16-s + (−1.27 + 7.23i)17-s + ⋯ |
L(s) = 1 | + (−0.696 − 0.122i)2-s + (0.738 + 0.674i)3-s + (0.469 + 0.171i)4-s + (0.975 − 0.219i)5-s + (−0.431 − 0.560i)6-s + (−1.39 + 0.806i)7-s + (−0.306 − 0.176i)8-s + (0.0893 + 0.996i)9-s + (−0.706 + 0.0332i)10-s + (−0.712 − 0.411i)11-s + (0.231 + 0.443i)12-s + (−0.762 + 0.639i)13-s + (1.07 − 0.390i)14-s + (0.868 + 0.496i)15-s + (0.191 + 0.160i)16-s + (−0.309 + 1.75i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.317 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.646023 + 0.897746i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.646023 + 0.897746i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.984 + 0.173i)T \) |
| 3 | \( 1 + (-1.27 - 1.16i)T \) |
| 5 | \( 1 + (-2.18 + 0.491i)T \) |
| 19 | \( 1 + (-0.726 - 4.29i)T \) |
good | 7 | \( 1 + (3.69 - 2.13i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.36 + 1.36i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.74 - 2.30i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.27 - 7.23i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (0.504 + 0.183i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (1.17 + 6.65i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-6.25 + 3.61i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 7.30T + 37T^{2} \) |
| 41 | \( 1 + (-4.84 - 4.06i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (0.172 + 0.473i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.399 + 2.26i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-0.105 + 0.291i)T + (-40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (1.70 - 9.66i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-1.45 - 0.530i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.976 - 5.53i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (2.79 - 1.01i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-8.13 + 9.69i)T + (-12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (9.01 - 10.7i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (7.26 + 12.5i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-7.27 + 6.10i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (0.308 - 1.75i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−10.43129341820185551893196641573, −9.961616645220952825383607941659, −9.403244577657301687035656473317, −8.614000400643872809628757786315, −7.81224749626496175801437865125, −6.28442921856019153664213714091, −5.78036985460453891736833769402, −4.20302787511442226826513505080, −2.86321120988743430599635293042, −2.12537782392352943894042235151,
0.67682509119439261158326630343, 2.53720563891818501276915150229, 3.03465868619927314863537122604, 5.01880997459938805166047587443, 6.37073356382297139662729452477, 7.07072585236014909427270198088, 7.54359510033508820560224828245, 8.932425869836501688607089336726, 9.662877857533773643442611593660, 10.00635045707473534370601318383