L(s) = 1 | + 2.23·5-s − 4.33i·7-s − 3·9-s − 7.50·17-s − 4.79i·23-s + 5.00·25-s − 8.78·29-s − 2.76i·31-s − 9.69i·35-s + 1.43·37-s − 12.7·41-s + 9.59i·43-s − 6.70·45-s − 11.7·49-s + 13.5·53-s + ⋯ |
L(s) = 1 | + 0.999·5-s − 1.63i·7-s − 9-s − 1.82·17-s − 0.999i·23-s + 1.00·25-s − 1.63·29-s − 0.496i·31-s − 1.63i·35-s + 0.236·37-s − 1.99·41-s + 1.46i·43-s − 0.999·45-s − 1.68·49-s + 1.86·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9403088643\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9403088643\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23T \) |
| 23 | \( 1 + 4.79iT \) |
good | 3 | \( 1 + 3T^{2} \) |
| 7 | \( 1 + 4.33iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 7.50T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 29 | \( 1 + 8.78T + 29T^{2} \) |
| 31 | \( 1 + 2.76iT - 31T^{2} \) |
| 37 | \( 1 - 1.43T + 37T^{2} \) |
| 41 | \( 1 + 12.7T + 41T^{2} \) |
| 43 | \( 1 - 9.59iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 13.5T + 53T^{2} \) |
| 59 | \( 1 - 4.16iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 11.1iT - 67T^{2} \) |
| 71 | \( 1 + 16.6iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 2.48iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 4.47T + 97T^{2} \) |
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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
−8.948219614264404708248828258699, −8.245110308070158140317053705378, −7.15528920482051563327362715960, −6.60122614804878787742450104004, −5.78863565717786573961320839524, −4.77141480057079528305573809145, −3.99344465131976454224075449409, −2.81794666121454283782123122131, −1.78237309686091182520245304873, −0.30772117322514140432805776825,
1.96457940962829690298066018271, 2.43061103322267684309924188724, 3.55728118899011783371262592912, 5.12080678932262965741698592465, 5.48814767882013500061529269119, 6.24140790015512100619186057071, 7.02133594978100657788180939045, 8.367101213699568166798683599962, 8.932287119261729372772344544319, 9.254922261210134813684190918475