L(s) = 1 | + 2.28i·3-s + 0.540·5-s + (−1.41 − 2.23i)7-s − 2.23·9-s − 1.23·11-s − 2.28·13-s + 1.23i·15-s − 1.74i·17-s + 5.11i·19-s + (5.11 − 3.23i)21-s − 3.23i·23-s − 4.70·25-s + 1.74i·27-s − 2i·29-s − 3.90·31-s + ⋯ |
L(s) = 1 | + 1.32i·3-s + 0.241·5-s + (−0.534 − 0.845i)7-s − 0.745·9-s − 0.372·11-s − 0.634·13-s + 0.319i·15-s − 0.423i·17-s + 1.17i·19-s + (1.11 − 0.706i)21-s − 0.674i·23-s − 0.941·25-s + 0.336i·27-s − 0.371i·29-s − 0.702·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.219 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.219 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3247704883\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3247704883\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (1.41 + 2.23i)T \) |
good | 3 | \( 1 - 2.28iT - 3T^{2} \) |
| 5 | \( 1 - 0.540T + 5T^{2} \) |
| 11 | \( 1 + 1.23T + 11T^{2} \) |
| 13 | \( 1 + 2.28T + 13T^{2} \) |
| 17 | \( 1 + 1.74iT - 17T^{2} \) |
| 19 | \( 1 - 5.11iT - 19T^{2} \) |
| 23 | \( 1 + 3.23iT - 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 + 3.90T + 31T^{2} \) |
| 37 | \( 1 + 6.94iT - 37T^{2} \) |
| 41 | \( 1 + 10.2iT - 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 + 8.47iT - 53T^{2} \) |
| 59 | \( 1 - 1.62iT - 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 - 3.70T + 67T^{2} \) |
| 71 | \( 1 - 10iT - 71T^{2} \) |
| 73 | \( 1 + 14.1iT - 73T^{2} \) |
| 79 | \( 1 + 3.52iT - 79T^{2} \) |
| 83 | \( 1 + 9.02iT - 83T^{2} \) |
| 89 | \( 1 - 15.4iT - 89T^{2} \) |
| 97 | \( 1 - 1.74iT - 97T^{2} \) |
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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
−9.381836732453351342916899814431, −8.375223842670019625872842178107, −7.49198875947839446657528562468, −6.66264076730303739490563844476, −5.58937648006929945433151256996, −4.91239546209864539789096290009, −3.93007444600780469926738286484, −3.44070265601804819170473780379, −2.07251972296706838714609929136, −0.11285961418822983061052578177,
1.52773350205451705773781405680, 2.41466936823798940392075617990, 3.27461694424472807644676365179, 4.81886400479449306922156310215, 5.63685284985343043737969254107, 6.50283932374320145028960784214, 6.98517677367743611730013265219, 7.939803694948330030085824051389, 8.502165515694890049308849687460, 9.565952396578279416083381481147