L(s) = 1 | + (−0.312 + 1.16i)3-s + (1.87 − 1.22i)5-s + (0.491 − 2.59i)7-s + (1.33 + 0.770i)9-s + (−1.67 − 2.90i)11-s + (2.92 + 2.92i)13-s + (0.840 + 2.56i)15-s + (0.259 + 0.0694i)17-s + (0.458 − 0.793i)19-s + (2.88 + 1.38i)21-s + (2.03 + 7.58i)23-s + (2.01 − 4.57i)25-s + (−3.87 + 3.87i)27-s − 1.31i·29-s + (3.25 − 1.88i)31-s + ⋯ |
L(s) = 1 | + (−0.180 + 0.673i)3-s + (0.837 − 0.546i)5-s + (0.185 − 0.982i)7-s + (0.444 + 0.256i)9-s + (−0.506 − 0.877i)11-s + (0.810 + 0.810i)13-s + (0.216 + 0.662i)15-s + (0.0628 + 0.0168i)17-s + (0.105 − 0.182i)19-s + (0.628 + 0.302i)21-s + (0.423 + 1.58i)23-s + (0.402 − 0.915i)25-s + (−0.746 + 0.746i)27-s − 0.244i·29-s + (0.584 − 0.337i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0328i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.44194 + 0.0236786i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.44194 + 0.0236786i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.87 + 1.22i)T \) |
| 7 | \( 1 + (-0.491 + 2.59i)T \) |
good | 3 | \( 1 + (0.312 - 1.16i)T + (-2.59 - 1.5i)T^{2} \) |
| 11 | \( 1 + (1.67 + 2.90i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.92 - 2.92i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.259 - 0.0694i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-0.458 + 0.793i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.03 - 7.58i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + 1.31iT - 29T^{2} \) |
| 31 | \( 1 + (-3.25 + 1.88i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (7.90 - 2.11i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 5.65iT - 41T^{2} \) |
| 43 | \( 1 + (2.61 - 2.61i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.244 + 0.911i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (13.0 + 3.49i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (3.91 + 6.78i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-9.70 - 5.60i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.44 - 5.37i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 + (1.96 - 7.33i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-1.87 - 1.08i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.49 - 5.49i)T + 83iT^{2} \) |
| 89 | \( 1 + (-1.34 + 2.33i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-10.1 + 10.1i)T - 97iT^{2} \) |
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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
−11.59722455226435036077764101240, −10.80187740768359599156616109540, −10.02918335510264394831890433384, −9.200593627149309366588670050321, −8.120951773571110912785235774765, −6.88333641156920453686688318905, −5.64005014958600857602559547948, −4.71097630533956691312037236932, −3.58155486914434700117791959600, −1.48112719068992087049016466174,
1.72166245029690899615315147737, 2.96046986241961106507894726562, 4.91406095368910820890140085886, 6.03350852161506002656412321328, 6.76471094325032693456982743929, 7.912121861658214703358437947706, 9.004099490045692189991033487570, 10.08281611086179999252107784966, 10.80602243748451199066458343641, 12.12117779645130189571506953544