L(s) = 1 | + (0.587 + 0.809i)2-s + (−1.73 + 0.564i)3-s + (−0.309 + 0.951i)4-s + (−1.47 − 1.07i)6-s + (−0.951 + 0.309i)8-s + (1.89 − 1.37i)9-s + (0.669 − 0.743i)11-s − 1.82i·12-s + (−0.809 − 0.587i)16-s + (1.14 − 1.58i)17-s + (2.22 + 0.722i)18-s + (−0.413 − 1.27i)19-s + (0.994 + 0.104i)22-s + (1.47 − 1.07i)24-s + (−1.43 + 1.97i)27-s + ⋯ |
L(s) = 1 | + (0.587 + 0.809i)2-s + (−1.73 + 0.564i)3-s + (−0.309 + 0.951i)4-s + (−1.47 − 1.07i)6-s + (−0.951 + 0.309i)8-s + (1.89 − 1.37i)9-s + (0.669 − 0.743i)11-s − 1.82i·12-s + (−0.809 − 0.587i)16-s + (1.14 − 1.58i)17-s + (2.22 + 0.722i)18-s + (−0.413 − 1.27i)19-s + (0.994 + 0.104i)22-s + (1.47 − 1.07i)24-s + (−1.43 + 1.97i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8175317389\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8175317389\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.587 - 0.809i)T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (-0.669 + 0.743i)T \) |
good | 3 | \( 1 + (1.73 - 0.564i)T + (0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (-1.14 + 1.58i)T + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (0.413 + 1.27i)T + (-0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.0646 + 0.198i)T + (-0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + 1.61iT - T^{2} \) |
| 47 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 - 1.95iT - T^{2} \) |
| 71 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (0.198 + 0.0646i)T + (0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (0.122 - 0.169i)T + (-0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + 1.82T + T^{2} \) |
| 97 | \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.314086758222820700458458723856, −8.593899455399907322614597201266, −7.14607170319770923810411897066, −7.00936555031588810781449076671, −5.88286213202844825658424645319, −5.53817375856005487963357887402, −4.72672673902626622138895116809, −4.03787136455879941832854757097, −2.98213242228761809014418750220, −0.68755379539879520625989763806,
1.25882526564517362465849320104, 1.87924576337131096830095890583, 3.61441022202505198757250286504, 4.39190837424731965774239335799, 5.21387686911990346734063703849, 6.10482989269260689782398277475, 6.27852538497729193451601267466, 7.37816242290749218117226240654, 8.309478385261598378443281380047, 9.683840389866109785422647645468