L(s) = 1 | + (−1.13 − 0.848i)2-s + (0.559 + 1.92i)4-s + 0.940i·5-s + 7-s + (0.996 − 2.64i)8-s + (0.798 − 1.06i)10-s + 5.98i·11-s − 6.59i·13-s + (−1.13 − 0.848i)14-s + (−3.37 + 2.14i)16-s + 2.64·17-s − 5.83i·19-s + (−1.80 + 0.526i)20-s + (5.08 − 6.77i)22-s − 2.88·23-s + ⋯ |
L(s) = 1 | + (−0.799 − 0.600i)2-s + (0.279 + 0.960i)4-s + 0.420i·5-s + 0.377·7-s + (0.352 − 0.935i)8-s + (0.252 − 0.336i)10-s + 1.80i·11-s − 1.82i·13-s + (−0.302 − 0.226i)14-s + (−0.843 + 0.537i)16-s + 0.640·17-s − 1.33i·19-s + (−0.403 + 0.117i)20-s + (1.08 − 1.44i)22-s − 0.601·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.199951764\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.199951764\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.13 + 0.848i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 - 0.940iT - 5T^{2} \) |
| 11 | \( 1 - 5.98iT - 11T^{2} \) |
| 13 | \( 1 + 6.59iT - 13T^{2} \) |
| 17 | \( 1 - 2.64T + 17T^{2} \) |
| 19 | \( 1 + 5.83iT - 19T^{2} \) |
| 23 | \( 1 + 2.88T + 23T^{2} \) |
| 29 | \( 1 + 3.09iT - 29T^{2} \) |
| 31 | \( 1 - 3.52T + 31T^{2} \) |
| 37 | \( 1 + 0.213iT - 37T^{2} \) |
| 41 | \( 1 - 1.63T + 41T^{2} \) |
| 43 | \( 1 - 7.16iT - 43T^{2} \) |
| 47 | \( 1 - 9.32T + 47T^{2} \) |
| 53 | \( 1 - 7.51iT - 53T^{2} \) |
| 59 | \( 1 + 11.9iT - 59T^{2} \) |
| 61 | \( 1 - 1.48iT - 61T^{2} \) |
| 67 | \( 1 - 13.0iT - 67T^{2} \) |
| 71 | \( 1 - 1.54T + 71T^{2} \) |
| 73 | \( 1 + 2.96T + 73T^{2} \) |
| 79 | \( 1 - 15.9T + 79T^{2} \) |
| 83 | \( 1 - 8.74iT - 83T^{2} \) |
| 89 | \( 1 + 7.50T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−9.665550675263673853721340622971, −8.679609744356778774825433142002, −7.73678039960585991182496513550, −7.41477768182220243414423488804, −6.42394019623897329845170474748, −5.12869402352727076149859832945, −4.25031583491596615901402691977, −2.99499166012987063996440946857, −2.30943307719463265523009320450, −0.889656630307940809558948251958,
0.917310500505801607638230396352, 1.98343660618740599398854094369, 3.55498233033461170668167776205, 4.67907313172413818968165540798, 5.69206158680716832559086112431, 6.24023750829072867883321126113, 7.20671325627383416087269958020, 8.114654301926572707502867249660, 8.699382490472803457602230680728, 9.206508380185648122553916171872