L(s) = 1 | + (−1.21 − 0.722i)2-s + (0.889 + 1.48i)3-s + (0.956 + 1.75i)4-s + (−0.00827 − 2.44i)6-s + (2.09 − 2.09i)7-s + (0.106 − 2.82i)8-s + (−1.41 + 2.64i)9-s + 2.65·11-s + (−1.75 + 2.98i)12-s + (4.21 − 4.21i)13-s + (−4.05 + 1.03i)14-s + (−2.17 + 3.35i)16-s + (−3.84 − 3.84i)17-s + (3.63 − 2.19i)18-s + 3.15·19-s + ⋯ |
L(s) = 1 | + (−0.859 − 0.510i)2-s + (0.513 + 0.857i)3-s + (0.478 + 0.878i)4-s + (−0.00337 − 0.999i)6-s + (0.790 − 0.790i)7-s + (0.0377 − 0.999i)8-s + (−0.472 + 0.881i)9-s + 0.800·11-s + (−0.507 + 0.861i)12-s + (1.16 − 1.16i)13-s + (−1.08 + 0.275i)14-s + (−0.542 + 0.839i)16-s + (−0.933 − 0.933i)17-s + (0.856 − 0.516i)18-s + 0.724·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.34526 - 0.102753i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.34526 - 0.102753i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.21 + 0.722i)T \) |
| 3 | \( 1 + (-0.889 - 1.48i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-2.09 + 2.09i)T - 7iT^{2} \) |
| 11 | \( 1 - 2.65T + 11T^{2} \) |
| 13 | \( 1 + (-4.21 + 4.21i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.84 + 3.84i)T + 17iT^{2} \) |
| 19 | \( 1 - 3.15T + 19T^{2} \) |
| 23 | \( 1 + (1.60 - 1.60i)T - 23iT^{2} \) |
| 29 | \( 1 + 4.35iT - 29T^{2} \) |
| 31 | \( 1 - 1.56T + 31T^{2} \) |
| 37 | \( 1 + (-4.94 - 4.94i)T + 37iT^{2} \) |
| 41 | \( 1 - 10.6iT - 41T^{2} \) |
| 43 | \( 1 + (-0.219 + 0.219i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.83 - 1.83i)T + 47iT^{2} \) |
| 53 | \( 1 + (-4.64 - 4.64i)T + 53iT^{2} \) |
| 59 | \( 1 + 9.93iT - 59T^{2} \) |
| 61 | \( 1 - 11.9iT - 61T^{2} \) |
| 67 | \( 1 + (8.80 + 8.80i)T + 67iT^{2} \) |
| 71 | \( 1 + 2.89iT - 71T^{2} \) |
| 73 | \( 1 + (2.29 + 2.29i)T + 73iT^{2} \) |
| 79 | \( 1 + 9.02iT - 79T^{2} \) |
| 83 | \( 1 + (-9.78 - 9.78i)T + 83iT^{2} \) |
| 89 | \( 1 - 2.83T + 89T^{2} \) |
| 97 | \( 1 + (9.26 - 9.26i)T - 97iT^{2} \) |
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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
−10.61034309130554691766526107962, −9.750442916439324679643367943500, −9.043475165603016059663457297629, −8.114165451766233397416679982729, −7.60538097731259219101927939721, −6.23898161598561716678796636100, −4.69619472438829811060456322038, −3.81398126868751573216768127715, −2.79846524902554607964759627276, −1.17731035557505737586030674185,
1.39807792133400836746255012275, 2.21510019314533504535942845634, 4.02087179784638136698578003456, 5.62617573428950777133736616565, 6.42932464849748955559605872774, 7.14934766548652212274241642535, 8.278205953178164022151612069261, 8.804029285096380962080948106768, 9.267891992828004324212098738609, 10.72618502697636886498961067506