L(s) = 1 | + (−0.340 + 1.26i)2-s + (−0.664 − 1.59i)3-s + (0.236 + 0.136i)4-s + (1.25 − 1.85i)5-s + (2.25 − 0.299i)6-s + (2.32 − 1.25i)7-s + (−2.11 + 2.11i)8-s + (−2.11 + 2.12i)9-s + (1.92 + 2.21i)10-s + (3.38 + 1.95i)11-s + (0.0611 − 0.468i)12-s + (−1.56 − 1.56i)13-s + (0.807 + 3.38i)14-s + (−3.79 − 0.767i)15-s + (−1.69 − 2.92i)16-s + (−2.58 + 0.693i)17-s + ⋯ |
L(s) = 1 | + (−0.240 + 0.897i)2-s + (−0.383 − 0.923i)3-s + (0.118 + 0.0681i)4-s + (0.559 − 0.829i)5-s + (0.921 − 0.122i)6-s + (0.879 − 0.476i)7-s + (−0.746 + 0.746i)8-s + (−0.705 + 0.708i)9-s + (0.609 + 0.701i)10-s + (1.01 + 0.588i)11-s + (0.0176 − 0.135i)12-s + (−0.434 − 0.434i)13-s + (0.215 + 0.903i)14-s + (−0.980 − 0.198i)15-s + (−0.422 − 0.731i)16-s + (−0.627 + 0.168i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.201i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 - 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.977819 + 0.0997019i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.977819 + 0.0997019i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.664 + 1.59i)T \) |
| 5 | \( 1 + (-1.25 + 1.85i)T \) |
| 7 | \( 1 + (-2.32 + 1.25i)T \) |
good | 2 | \( 1 + (0.340 - 1.26i)T + (-1.73 - i)T^{2} \) |
| 11 | \( 1 + (-3.38 - 1.95i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.56 + 1.56i)T + 13iT^{2} \) |
| 17 | \( 1 + (2.58 - 0.693i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (1.61 - 0.930i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.38 + 0.638i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 - 0.513T + 29T^{2} \) |
| 31 | \( 1 + (4.29 - 7.43i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (6.60 + 1.77i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 0.308iT - 41T^{2} \) |
| 43 | \( 1 + (-7.60 - 7.60i)T + 43iT^{2} \) |
| 47 | \( 1 + (1.36 - 5.10i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.498 - 1.85i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-0.259 + 0.448i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.55 + 4.42i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.34 - 8.74i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 15.3iT - 71T^{2} \) |
| 73 | \( 1 + (-2.79 + 0.749i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-4.37 + 2.52i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-9.16 + 9.16i)T - 83iT^{2} \) |
| 89 | \( 1 + (5.67 + 9.82i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (6.81 - 6.81i)T - 97iT^{2} \) |
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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
−14.00467130269695573303423589526, −12.65220094476052329926805090928, −11.97811278527315043932502869841, −10.76990987310771620923462650517, −9.035451114484931641869710248700, −8.083905201397075981553730335517, −7.10607630303252574186231754349, −6.06944224308059868520903317207, −4.84054053269097728575536111184, −1.82756279041259597197516985663,
2.24905959127290116076869027754, 3.87371796419549271614069142140, 5.64653931949863025770902599871, 6.72755079902891027125056980273, 8.885571103154160664878071026267, 9.672886849462221488776597390094, 10.78576093012541445434214948499, 11.34664175645738743097367701244, 12.09971993693210005276973881984, 13.93215871242626676792513798694