L(s) = 1 | + (−0.819 − 0.573i)2-s + (−0.0360 + 1.73i)3-s + (0.342 + 0.939i)4-s + (0.368 − 2.20i)5-s + (1.02 − 1.39i)6-s + (−0.623 + 0.166i)7-s + (0.258 − 0.965i)8-s + (−2.99 − 0.125i)9-s + (−1.56 + 1.59i)10-s + (−1.95 + 1.12i)11-s + (−1.63 + 0.558i)12-s + (−3.88 + 0.340i)13-s + (0.606 + 0.220i)14-s + (3.80 + 0.717i)15-s + (−0.766 + 0.642i)16-s + (−4.32 − 3.03i)17-s + ⋯ |
L(s) = 1 | + (−0.579 − 0.405i)2-s + (−0.0208 + 0.999i)3-s + (0.171 + 0.469i)4-s + (0.164 − 0.986i)5-s + (0.417 − 0.570i)6-s + (−0.235 + 0.0630i)7-s + (0.0915 − 0.341i)8-s + (−0.999 − 0.0416i)9-s + (−0.495 + 0.504i)10-s + (−0.588 + 0.339i)11-s + (−0.473 + 0.161i)12-s + (−1.07 + 0.0943i)13-s + (0.161 + 0.0589i)14-s + (0.982 + 0.185i)15-s + (−0.191 + 0.160i)16-s + (−1.04 − 0.735i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00144490 + 0.0612468i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00144490 + 0.0612468i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.819 + 0.573i)T \) |
| 3 | \( 1 + (0.0360 - 1.73i)T \) |
| 5 | \( 1 + (-0.368 + 2.20i)T \) |
| 19 | \( 1 + (-0.0260 - 4.35i)T \) |
good | 7 | \( 1 + (0.623 - 0.166i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (1.95 - 1.12i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.88 - 0.340i)T + (12.8 - 2.25i)T^{2} \) |
| 17 | \( 1 + (4.32 + 3.03i)T + (5.81 + 15.9i)T^{2} \) |
| 23 | \( 1 + (-1.63 - 0.760i)T + (14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (1.38 - 7.84i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.69 + 2.93i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.860 - 0.860i)T + 37iT^{2} \) |
| 41 | \( 1 + (3.79 + 4.52i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (8.64 - 4.02i)T + (27.6 - 32.9i)T^{2} \) |
| 47 | \( 1 + (6.88 + 9.82i)T + (-16.0 + 44.1i)T^{2} \) |
| 53 | \( 1 + (6.10 + 2.84i)T + (34.0 + 40.6i)T^{2} \) |
| 59 | \( 1 + (0.503 + 2.85i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-9.92 + 3.61i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (0.903 - 0.632i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (-0.913 + 2.51i)T + (-54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (0.670 - 7.65i)T + (-71.8 - 12.6i)T^{2} \) |
| 79 | \( 1 + (-6.38 - 7.60i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-2.78 + 0.746i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (-8.24 - 6.92i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-7.15 + 10.2i)T + (-33.1 - 91.1i)T^{2} \) |
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
−11.05032978028888446790166079140, −9.928405745377217840652247294981, −9.678306262813770178143454796369, −8.745241171300655017033694082840, −8.008275747518589920798988043615, −6.74729375497551975852578093495, −5.26036066413126972934322332015, −4.72209216112666115654721494242, −3.42438627486341817827923550605, −2.10673072168970170237728063365,
0.03815841844299119662337670408, 2.09730983242718478436404747433, 3.00318801748458050044804215915, 4.94831624648352777927575238243, 6.22054015649811320238334188513, 6.70249764306347439728531965279, 7.57050288469515684453628376199, 8.294273962890328365971599190546, 9.377280552808140185372641528723, 10.31497887448004469139554502226