| L(s) = 1 | − 2·7-s − 13.8i·13-s + (−13 + 13.8i)19-s − 25·25-s − 41.5i·31-s − 69.2i·37-s − 22·43-s − 45·49-s − 74·61-s − 55.4i·67-s − 46·73-s − 69.2i·79-s + 27.7i·91-s − 193. i·97-s + 69.2i·103-s + ⋯ |
| L(s) = 1 | − 0.285·7-s − 1.06i·13-s + (−0.684 + 0.729i)19-s − 25-s − 1.34i·31-s − 1.87i·37-s − 0.511·43-s − 0.918·49-s − 1.21·61-s − 0.827i·67-s − 0.630·73-s − 0.876i·79-s + 0.304i·91-s − 1.99i·97-s + 0.672i·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.684 + 0.729i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.684 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7469556449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7469556449\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (13 - 13.8i)T \) |
| good | 5 | \( 1 + 25T^{2} \) |
| 7 | \( 1 + 2T + 49T^{2} \) |
| 11 | \( 1 + 121T^{2} \) |
| 13 | \( 1 + 13.8iT - 169T^{2} \) |
| 17 | \( 1 + 289T^{2} \) |
| 23 | \( 1 + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 + 41.5iT - 961T^{2} \) |
| 37 | \( 1 + 69.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 + 22T + 1.84e3T^{2} \) |
| 47 | \( 1 + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 + 74T + 3.72e3T^{2} \) |
| 67 | \( 1 + 55.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 46T + 5.32e3T^{2} \) |
| 79 | \( 1 + 69.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 + 193. iT - 9.40e3T^{2} \) |
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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
−10.00229317590574280893089601781, −9.169612419411650410219666604447, −8.106805326885983890349195914311, −7.49226122326568018692123791774, −6.22494029760490995195735330313, −5.59418225127288183103015524993, −4.29757220011257812927228317226, −3.29908411790347615167131401283, −1.99768614249191716496183668197, −0.25647752265183202834347007699,
1.60954175719711653424967202922, 2.94008588727046206248111386239, 4.15052676258395661547726760607, 5.06501410485378818501030534332, 6.34757415330698108388573505031, 6.90483454677010773843409982406, 8.074341752095899993765634886160, 8.918602733232598266644600421781, 9.703544158313257447368248180357, 10.55882131995040134463913996220
