L(s) = 1 | + (−1.32 − 1.11i)3-s − 2·4-s − 2.23i·5-s − 2.64·7-s + (0.5 + 2.95i)9-s − 5.91i·11-s + (2.64 + 2.23i)12-s + 2.64·13-s + (−2.50 + 2.95i)15-s + 4·16-s + 2.23i·17-s + 4.47i·20-s + (3.50 + 2.95i)21-s − 5.00·25-s + (2.64 − 4.47i)27-s + 5.29·28-s + ⋯ |
L(s) = 1 | + (−0.763 − 0.645i)3-s − 4-s − 0.999i·5-s − 0.999·7-s + (0.166 + 0.986i)9-s − 1.78i·11-s + (0.763 + 0.645i)12-s + 0.733·13-s + (−0.645 + 0.763i)15-s + 16-s + 0.542i·17-s + 0.999i·20-s + (0.763 + 0.645i)21-s − 1.00·25-s + (0.509 − 0.860i)27-s + 0.999·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.645 + 0.763i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.645 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.210750 - 0.454054i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.210750 - 0.454054i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.32 + 1.11i)T \) |
| 5 | \( 1 + 2.23iT \) |
| 7 | \( 1 + 2.64T \) |
good | 2 | \( 1 + 2T^{2} \) |
| 11 | \( 1 + 5.91iT - 11T^{2} \) |
| 13 | \( 1 - 2.64T + 13T^{2} \) |
| 17 | \( 1 - 2.23iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 5.91iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 11.1iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 11.8iT - 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 + T + 79T^{2} \) |
| 83 | \( 1 - 8.94iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 18.5T + 97T^{2} \) |
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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
−13.34741294708740758107369255322, −12.60586758942157858160174748866, −11.47685159915176114164733461736, −10.17388558710500914193221020720, −8.870322178775433819924699679718, −8.123299798078636550615703189610, −6.21577155880049371115133005822, −5.45555262408562740704452774358, −3.82719619637920896474875253525, −0.66368167937209263439652868992,
3.47110627331364208458537828722, 4.71280274439209525804059574563, 6.15362876635385216636892125316, 7.26360276363798949662648662762, 9.224795699877389279750112784389, 9.897870467817826874600174401912, 10.73282313894775253261756710319, 12.13744092956451481461377199722, 12.99168797998621854376252451316, 14.25893519281782353489141038097