L(s) = 1 | + (1.5 − 0.866i)3-s − 3·5-s + (2.5 − 0.866i)7-s + (1.5 − 2.59i)9-s + 1.73i·11-s + (−1.5 − 0.866i)13-s + (−4.5 + 2.59i)15-s + (1.5 − 2.59i)17-s + (4.5 − 2.59i)19-s + (3 − 3.46i)21-s − 5.19i·23-s + 4·25-s − 5.19i·27-s + (−4.5 + 2.59i)29-s + (3 − 1.73i)31-s + ⋯ |
L(s) = 1 | + (0.866 − 0.499i)3-s − 1.34·5-s + (0.944 − 0.327i)7-s + (0.5 − 0.866i)9-s + 0.522i·11-s + (−0.416 − 0.240i)13-s + (−1.16 + 0.670i)15-s + (0.363 − 0.630i)17-s + (1.03 − 0.596i)19-s + (0.654 − 0.755i)21-s − 1.08i·23-s + 0.800·25-s − 0.999i·27-s + (−0.835 + 0.482i)29-s + (0.538 − 0.311i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.766710297\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.766710297\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 + 0.866i)T \) |
| 7 | \( 1 + (-2.5 + 0.866i)T \) |
good | 5 | \( 1 + 3T + 5T^{2} \) |
| 11 | \( 1 - 1.73iT - 11T^{2} \) |
| 13 | \( 1 + (1.5 + 0.866i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.5 + 2.59i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 5.19iT - 23T^{2} \) |
| 29 | \( 1 + (4.5 - 2.59i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3 + 1.73i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (7.5 + 4.33i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-12 - 6.92i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 3.46iT - 71T^{2} \) |
| 73 | \( 1 + (4.5 + 2.59i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.5 - 12.9i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.5 - 2.59i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.5 - 0.866i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.604973725224556736570004592838, −8.757784068156949570530425481935, −7.896357013943316189036757352021, −7.49044637993106986471876771949, −6.86518808339671049873863321449, −5.19583477218089086782252856814, −4.32202318720370620718751695799, −3.44177602631156236998996209228, −2.29935325774397855562173573792, −0.78411590172673090629700142785,
1.60219997593976960311309214279, 3.09933884996784626575563988469, 3.83009221647176563415124509143, 4.71800139539361215846507755566, 5.63359605617132938065967842633, 7.21474326828636444407408044873, 7.956753267538786062637761634045, 8.235089002227363815771396552624, 9.237693193369405064129562653205, 10.06471069768450256329849144386