L(s) = 1 | + (−0.481 − 1.66i)3-s + (−0.160 − 1.99i)4-s + (−3.64 + 1.46i)7-s + (−2.53 + 1.60i)9-s + (−3.23 + 1.22i)12-s + (−2.59 + 2.5i)13-s + (−3.94 + 0.641i)16-s + (5.49 − 1.47i)19-s + (4.19 + 5.35i)21-s + (−3.31 − 3.74i)25-s + (3.88 + 3.44i)27-s + (3.51 + 7.03i)28-s + (0.438 + 7.24i)31-s + (3.60 + 4.79i)36-s + (−6.69 − 10.1i)37-s + ⋯ |
L(s) = 1 | + (−0.278 − 0.960i)3-s + (−0.0804 − 0.996i)4-s + (−1.37 + 0.554i)7-s + (−0.845 + 0.534i)9-s + (−0.935 + 0.354i)12-s + (−0.720 + 0.693i)13-s + (−0.987 + 0.160i)16-s + (1.26 − 0.337i)19-s + (0.915 + 1.16i)21-s + (−0.663 − 0.748i)25-s + (0.748 + 0.663i)27-s + (0.663 + 1.32i)28-s + (0.0786 + 1.30i)31-s + (0.600 + 0.799i)36-s + (−1.10 − 1.66i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.652 - 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.652 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0698747 + 0.152459i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0698747 + 0.152459i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.481 + 1.66i)T \) |
| 13 | \( 1 + (2.59 - 2.5i)T \) |
good | 2 | \( 1 + (0.160 + 1.99i)T^{2} \) |
| 5 | \( 1 + (3.31 + 3.74i)T^{2} \) |
| 7 | \( 1 + (3.64 - 1.46i)T + (5.04 - 4.84i)T^{2} \) |
| 11 | \( 1 + (-9.93 + 4.71i)T^{2} \) |
| 17 | \( 1 + (11.7 + 12.2i)T^{2} \) |
| 19 | \( 1 + (-5.49 + 1.47i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (28.9 - 2.33i)T^{2} \) |
| 31 | \( 1 + (-0.438 - 7.24i)T + (-30.7 + 3.73i)T^{2} \) |
| 37 | \( 1 + (6.69 + 10.1i)T + (-14.5 + 34.0i)T^{2} \) |
| 41 | \( 1 + (21.9 + 34.6i)T^{2} \) |
| 43 | \( 1 + (12.4 - 2.53i)T + (39.5 - 16.8i)T^{2} \) |
| 47 | \( 1 + (43.9 - 16.6i)T^{2} \) |
| 53 | \( 1 + (51.4 - 12.6i)T^{2} \) |
| 59 | \( 1 + (-18.6 + 55.9i)T^{2} \) |
| 61 | \( 1 + (12.4 + 9.38i)T + (16.9 + 58.5i)T^{2} \) |
| 67 | \( 1 + (6.63 + 7.80i)T + (-10.7 + 66.1i)T^{2} \) |
| 71 | \( 1 + (-70.9 + 2.85i)T^{2} \) |
| 73 | \( 1 + (1.60 - 5.15i)T + (-60.0 - 41.4i)T^{2} \) |
| 79 | \( 1 + (8.45 - 12.2i)T + (-28.0 - 73.8i)T^{2} \) |
| 83 | \( 1 + (-38.5 - 73.4i)T^{2} \) |
| 89 | \( 1 + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-1.70 + 16.9i)T + (-95.0 - 19.4i)T^{2} \) |
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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
−10.23432719980748448688014846121, −9.509714092026067525382359702299, −8.726620807134414795493618051986, −7.27568816860675624004811438676, −6.61245367402334725176013945411, −5.82474195603312589455397281499, −4.94322852684127128953648988569, −3.08767748715571094108039965021, −1.83616184795605944486403103495, −0.097337389466126812515423586290,
3.06778052177355812319489869608, 3.52632418158682961006849401652, 4.71098762333120817415232483426, 5.85977898410385106289546796707, 6.99589469644383244489053920336, 7.85668273828177456663919338978, 9.016474076465623049154031638502, 9.835000635003275579012786671995, 10.29680262719255646570341832920, 11.65163634753640032956255229513