L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (1.22 − 2.12i)5-s + (2.62 − 0.358i)7-s − 0.999i·8-s − 2.44i·10-s + (2.12 + 1.22i)13-s + (2.09 − 1.62i)14-s + (−0.5 − 0.866i)16-s + 4.89·17-s + 2.44i·19-s + (−1.22 − 2.12i)20-s + (−5.19 − 3i)23-s + (−0.499 − 0.866i)25-s + 2.44·26-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.547 − 0.948i)5-s + (0.990 − 0.135i)7-s − 0.353i·8-s − 0.774i·10-s + (0.588 + 0.339i)13-s + (0.558 − 0.433i)14-s + (−0.125 − 0.216i)16-s + 1.18·17-s + 0.561i·19-s + (−0.273 − 0.474i)20-s + (−1.08 − 0.625i)23-s + (−0.0999 − 0.173i)25-s + 0.480·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.908638251\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.908638251\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.62 + 0.358i)T \) |
good | 5 | \( 1 + (-1.22 + 2.12i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.12 - 1.22i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 - 2.44iT - 19T^{2} \) |
| 23 | \( 1 + (5.19 + 3i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (5.19 - 3i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + (-2.44 + 4.24i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (2.44 + 4.24i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + (-6.12 + 10.6i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (10.6 - 6.12i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 9.79iT - 73T^{2} \) |
| 79 | \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.22 + 2.12i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + (-4.24 + 2.44i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.769357135473954000554865009912, −8.818938775935796422400993529116, −8.135862817999838690820243729097, −7.14017168694153478789500924376, −5.83572625053274000966903089940, −5.40418595347673251957297976516, −4.44303032388492257212804518759, −3.59456103539812716258165621470, −2.01573629991917285874962258723, −1.22386238742311741925763800420,
1.69524951540088176943988835088, 2.84508913463090600907026690123, 3.81277482729393081722216660919, 4.96845246060683861195337867616, 5.81639725737148039313957639042, 6.41656185657135664579191662677, 7.60691181511234430142737200438, 7.959863210642286428219130393439, 9.152956091169454803305630234157, 10.11533208545346331722955978732