L(s) = 1 | + (−0.258 − 0.965i)3-s + (−2.36 + 1.18i)7-s + (−0.866 + 0.499i)9-s + (−2.25 + 3.90i)11-s + (4.37 − 4.37i)13-s + (4.35 − 1.16i)17-s + (−2.51 − 4.34i)19-s + (1.75 + 1.97i)21-s + (−1.44 + 5.40i)23-s + (0.707 + 0.707i)27-s + (−3.92 − 2.26i)31-s + (4.35 + 1.16i)33-s + (−1.67 − 0.448i)37-s + (−5.35 − 3.09i)39-s − 2.22i·41-s + ⋯ |
L(s) = 1 | + (−0.149 − 0.557i)3-s + (−0.894 + 0.447i)7-s + (−0.288 + 0.166i)9-s + (−0.680 + 1.17i)11-s + (1.21 − 1.21i)13-s + (1.05 − 0.283i)17-s + (−0.575 − 0.997i)19-s + (0.383 + 0.431i)21-s + (−0.301 + 1.12i)23-s + (0.136 + 0.136i)27-s + (−0.704 − 0.406i)31-s + (0.758 + 0.203i)33-s + (−0.275 − 0.0736i)37-s + (−0.857 − 0.495i)39-s − 0.347i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.389 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.389 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.275180341\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.275180341\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.258 + 0.965i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.36 - 1.18i)T \) |
good | 11 | \( 1 + (2.25 - 3.90i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.37 + 4.37i)T - 13iT^{2} \) |
| 17 | \( 1 + (-4.35 + 1.16i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (2.51 + 4.34i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.44 - 5.40i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + (3.92 + 2.26i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.67 + 0.448i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 2.22iT - 41T^{2} \) |
| 43 | \( 1 + (-7.85 - 7.85i)T + 43iT^{2} \) |
| 47 | \( 1 + (-3.33 + 12.4i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-11.8 + 3.17i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (1.11 - 1.92i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.34 + 3.66i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.29 + 4.84i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 + (1.12 + 4.19i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (1.58 - 0.914i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-11.2 + 11.2i)T - 83iT^{2} \) |
| 89 | \( 1 + (3.90 + 6.76i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.53 - 3.53i)T + 97iT^{2} \) |
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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
−8.959947881156297084971305356384, −8.061027769956146353832035030345, −7.40361310881196424836517271094, −6.66902721531675535920713489903, −5.66868189212180049185736204863, −5.31964126097045439437635488586, −3.87574297283185807733844336595, −2.99493338041618853592287558603, −2.04092892463176403921910116519, −0.57042926753180027129779365360,
0.991212315566463352097468986377, 2.60083140432316627601361952915, 3.78683125295657863927759951962, 3.96815443248011076101452421635, 5.44935052320356627948927462392, 6.05773827096086896466747647874, 6.69284523929644926360759418189, 7.83157539365231574023989193168, 8.601172076846469521100164385900, 9.186176522620874142316095171361