L(s) = 1 | + (−1.75 + 0.961i)2-s + (0.903 + 2.86i)3-s + (2.15 − 3.37i)4-s + (4.95 + 0.663i)5-s + (−4.33 − 4.14i)6-s + (−7.30 + 7.30i)7-s + (−0.535 + 7.98i)8-s + (−7.36 + 5.17i)9-s + (−9.32 + 3.59i)10-s + 4.41·11-s + (11.5 + 3.11i)12-s + (7.53 − 7.53i)13-s + (5.78 − 19.8i)14-s + (2.58 + 14.7i)15-s + (−6.73 − 14.5i)16-s + (−0.350 + 0.350i)17-s + ⋯ |
L(s) = 1 | + (−0.876 + 0.480i)2-s + (0.301 + 0.953i)3-s + (0.538 − 0.842i)4-s + (0.991 + 0.132i)5-s + (−0.722 − 0.691i)6-s + (−1.04 + 1.04i)7-s + (−0.0669 + 0.997i)8-s + (−0.818 + 0.574i)9-s + (−0.932 + 0.359i)10-s + 0.401·11-s + (0.965 + 0.259i)12-s + (0.579 − 0.579i)13-s + (0.413 − 1.41i)14-s + (0.172 + 0.985i)15-s + (−0.420 − 0.907i)16-s + (−0.0206 + 0.0206i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.157 - 0.987i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.157 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.602983 + 0.706939i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.602983 + 0.706939i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.75 - 0.961i)T \) |
| 3 | \( 1 + (-0.903 - 2.86i)T \) |
| 5 | \( 1 + (-4.95 - 0.663i)T \) |
good | 7 | \( 1 + (7.30 - 7.30i)T - 49iT^{2} \) |
| 11 | \( 1 - 4.41T + 121T^{2} \) |
| 13 | \( 1 + (-7.53 + 7.53i)T - 169iT^{2} \) |
| 17 | \( 1 + (0.350 - 0.350i)T - 289iT^{2} \) |
| 19 | \( 1 - 9.24T + 361T^{2} \) |
| 23 | \( 1 + (-17.9 + 17.9i)T - 529iT^{2} \) |
| 29 | \( 1 - 5.52T + 841T^{2} \) |
| 31 | \( 1 + 48.1iT - 961T^{2} \) |
| 37 | \( 1 + (-3.39 - 3.39i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 33.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (-1.45 - 1.45i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (27.8 + 27.8i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (52.6 + 52.6i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 - 24.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 46.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + (32.1 - 32.1i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 116.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (72.2 - 72.2i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 55.9T + 6.24e3T^{2} \) |
| 83 | \( 1 + (46.5 - 46.5i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 33.2T + 7.92e3T^{2} \) |
| 97 | \( 1 + (24.6 + 24.6i)T + 9.40e3iT^{2} \) |
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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
−15.33015406464679778388060598886, −14.48253723762563881538272553663, −13.12441848010360755528814782641, −11.31128724924810814049185129729, −10.03785793334837352169504010770, −9.410127644563392754380851798174, −8.477347057836466198876170883750, −6.41705630496557290650920773404, −5.46122981099885893695941770774, −2.75504241564119711309187992230,
1.31585993617653015168059582112, 3.26485853351258955966322827421, 6.40296220975341106618860511670, 7.21986352445130882492419696760, 8.855164866139588679555367008291, 9.705764532318983992287685610718, 10.97715623158191874962860734231, 12.40243795602655018462656548002, 13.31575660838173715613978655490, 14.05656377703685320196852076223