L(s) = 1 | + (−1.67 − 0.451i)3-s − 0.203·5-s + (−1.27 + 2.32i)7-s + (2.59 + 1.50i)9-s − 4.46i·11-s + (1.25 − 0.725i)13-s + (0.339 + 0.0916i)15-s + (−1.60 − 2.78i)17-s + (−6.20 − 3.58i)19-s + (3.17 − 3.30i)21-s + 1.26i·23-s − 4.95·25-s + (−3.65 − 3.69i)27-s + (−0.944 − 0.545i)29-s + (−5.60 − 3.23i)31-s + ⋯ |
L(s) = 1 | + (−0.965 − 0.260i)3-s − 0.0908·5-s + (−0.480 + 0.876i)7-s + (0.864 + 0.502i)9-s − 1.34i·11-s + (0.348 − 0.201i)13-s + (0.0877 + 0.0236i)15-s + (−0.389 − 0.674i)17-s + (−1.42 − 0.821i)19-s + (0.692 − 0.721i)21-s + 0.264i·23-s − 0.991·25-s + (−0.703 − 0.710i)27-s + (−0.175 − 0.101i)29-s + (−1.00 − 0.580i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.721 + 0.692i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.721 + 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.159292 - 0.395942i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.159292 - 0.395942i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.67 + 0.451i)T \) |
| 7 | \( 1 + (1.27 - 2.32i)T \) |
good | 5 | \( 1 + 0.203T + 5T^{2} \) |
| 11 | \( 1 + 4.46iT - 11T^{2} \) |
| 13 | \( 1 + (-1.25 + 0.725i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.60 + 2.78i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (6.20 + 3.58i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 1.26iT - 23T^{2} \) |
| 29 | \( 1 + (0.944 + 0.545i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (5.60 + 3.23i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.02 + 5.24i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.370 + 0.642i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.69 - 8.13i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.0465 + 0.0806i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-9.35 + 5.39i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.16 + 8.94i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (7.34 - 4.24i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.02 + 6.97i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 15.6iT - 71T^{2} \) |
| 73 | \( 1 + (-0.984 + 0.568i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.86 - 10.1i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.29 - 3.98i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (3.52 - 6.10i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.17 + 1.83i)T + (48.5 + 84.0i)T^{2} \) |
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
−11.00954426854454471824545265779, −9.697050296341691657104248683110, −8.837884412293979364543961200630, −7.85970336976971159661941830106, −6.60181168008833035498849998564, −5.99371328336773374406540297016, −5.14067307856231260658577162640, −3.76710634900833574759164648030, −2.28079154617139180035498240862, −0.27884175851626709948874697279,
1.73998378726228518387701472187, 3.89550756076535965142239688125, 4.39647838782050353404291764460, 5.75764916383064869988187118813, 6.67829083068988980223548477651, 7.34158214395125813882159945444, 8.638368631258759300170194450833, 9.893550563486968484812938769295, 10.31319168115497423571597432636, 11.10740707165728721136840690871