| L(s) = 1 | − 2.30i·2-s − 3.30·4-s + 4.30i·7-s + 3.00i·8-s + 11-s − 5i·13-s + 9.90·14-s + 0.302·16-s + 3.90i·17-s + 19-s − 2.30i·22-s − 3.69i·23-s − 11.5·26-s − 14.2i·28-s − 9.90·29-s + ⋯ |
| L(s) = 1 | − 1.62i·2-s − 1.65·4-s + 1.62i·7-s + 1.06i·8-s + 0.301·11-s − 1.38i·13-s + 2.64·14-s + 0.0756·16-s + 0.947i·17-s + 0.229·19-s − 0.490i·22-s − 0.770i·23-s − 2.25·26-s − 2.68i·28-s − 1.83·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7240788996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7240788996\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 2.30iT - 2T^{2} \) |
| 7 | \( 1 - 4.30iT - 7T^{2} \) |
| 13 | \( 1 + 5iT - 13T^{2} \) |
| 17 | \( 1 - 3.90iT - 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 + 3.69iT - 23T^{2} \) |
| 29 | \( 1 + 9.90T + 29T^{2} \) |
| 31 | \( 1 + 4.21T + 31T^{2} \) |
| 37 | \( 1 - 9.60iT - 37T^{2} \) |
| 41 | \( 1 + 1.60T + 41T^{2} \) |
| 43 | \( 1 - 7.21iT - 43T^{2} \) |
| 47 | \( 1 - 3iT - 47T^{2} \) |
| 53 | \( 1 - 2.30iT - 53T^{2} \) |
| 59 | \( 1 - 0.211T + 59T^{2} \) |
| 61 | \( 1 - 2.90T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 4.60T + 71T^{2} \) |
| 73 | \( 1 + 2.90iT - 73T^{2} \) |
| 79 | \( 1 - 0.0916T + 79T^{2} \) |
| 83 | \( 1 - 14.5iT - 83T^{2} \) |
| 89 | \( 1 - 5.30T + 89T^{2} \) |
| 97 | \( 1 - 11.6iT - 97T^{2} \) |
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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
−9.217259409140272331267150728014, −8.509457382060831380819447523383, −7.85570132406270318544265835514, −6.44113167439480252746020990702, −5.63924390106091339059433423685, −4.93732343493831845446447120603, −3.79149923119794081943099879131, −3.03574306006582267903557845198, −2.30286905979364061834663335628, −1.35390402253217914796483478653,
0.24524441011154242126091831906, 1.81273696238530868919186253205, 3.70724272660565102095583678388, 4.17427284963408883837139389623, 5.11921101661834185120088690860, 5.87373205691039388007786322652, 6.95376804940651988283374463904, 7.19185816788591160462673414752, 7.64823460884670578523399290131, 8.839290891079941246430633716413
