L(s) = 1 | + (1.73 − i)2-s + (0.866 + 0.5i)3-s + (0.999 − 1.73i)4-s + (0.133 + 2.23i)5-s + 1.99·6-s + (−1 − 1.73i)9-s + (2.46 + 3.73i)10-s + (1.5 − 2.59i)11-s + (1.73 − i)12-s − i·13-s + (−1 + 1.99i)15-s + (1.99 + 3.46i)16-s + (−6.06 − 3.5i)17-s + (−3.46 − 2i)18-s + (3.99 + 1.99i)20-s + ⋯ |
L(s) = 1 | + (1.22 − 0.707i)2-s + (0.499 + 0.288i)3-s + (0.499 − 0.866i)4-s + (0.0599 + 0.998i)5-s + 0.816·6-s + (−0.333 − 0.577i)9-s + (0.779 + 1.18i)10-s + (0.452 − 0.783i)11-s + (0.499 − 0.288i)12-s − 0.277i·13-s + (−0.258 + 0.516i)15-s + (0.499 + 0.866i)16-s + (−1.47 − 0.848i)17-s + (−0.816 − 0.471i)18-s + (0.894 + 0.447i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.46100 - 0.418518i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.46100 - 0.418518i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.133 - 2.23i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-1.73 + i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.866 - 0.5i)T + (1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 + (6.06 + 3.5i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (5.19 - 3i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.73 + i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + (2.59 - 1.5i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.19 - 3i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5 + 8.66i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4 - 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.73 + i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + (-5.19 - 3i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.5 + 4.33i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 7iT - 97T^{2} \) |
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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
−11.84734073445615325784304562920, −11.40457548371683676834023638938, −10.45137601362714446281201546191, −9.307299177315167563285402299268, −8.211024705438738982717088067916, −6.66794379850821454051260255062, −5.77936401919798097141662208653, −4.27166356562653019724860499909, −3.34073641953695866745248311369, −2.47770620534856526771575868493,
2.12726729207640988209960524677, 4.05571856441192226551206327509, 4.75157297596632432613567126781, 5.94195644595370077774356314881, 6.94072674383679909940760900255, 8.106984326874873052146191642680, 8.931265494229016992638800980414, 10.16057695017574218661560667723, 11.66834862359525183410145711982, 12.57462421800946410490802857583