| L(s) = 1 | + (0.239 + 3.20i)3-s + (−0.131 + 0.0405i)5-s + (−0.934 + 1.61i)7-s + (−7.22 + 1.08i)9-s + (−1.63 − 2.05i)11-s + (1.27 + 1.18i)13-s + (−0.161 − 0.411i)15-s + (1.45 + 0.448i)17-s + (−4.83 − 0.729i)19-s + (−5.40 − 2.60i)21-s + (−1.59 + 4.07i)23-s + (−4.11 + 2.80i)25-s + (−3.07 − 13.4i)27-s + (−0.00236 + 0.0315i)29-s + (2.33 + 1.59i)31-s + ⋯ |
| L(s) = 1 | + (0.138 + 1.84i)3-s + (−0.0588 + 0.0181i)5-s + (−0.353 + 0.611i)7-s + (−2.40 + 0.362i)9-s + (−0.493 − 0.618i)11-s + (0.354 + 0.329i)13-s + (−0.0416 − 0.106i)15-s + (0.352 + 0.108i)17-s + (−1.10 − 0.167i)19-s + (−1.17 − 0.568i)21-s + (−0.333 + 0.848i)23-s + (−0.823 + 0.561i)25-s + (−0.591 − 2.59i)27-s + (−0.000438 + 0.00585i)29-s + (0.420 + 0.286i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.302i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.953 + 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.139795 - 0.903003i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.139795 - 0.903003i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 43 | \( 1 + (-5.73 + 3.17i)T \) |
| good | 3 | \( 1 + (-0.239 - 3.20i)T + (-2.96 + 0.447i)T^{2} \) |
| 5 | \( 1 + (0.131 - 0.0405i)T + (4.13 - 2.81i)T^{2} \) |
| 7 | \( 1 + (0.934 - 1.61i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.63 + 2.05i)T + (-2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-1.27 - 1.18i)T + (0.971 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-1.45 - 0.448i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (4.83 + 0.729i)T + (18.1 + 5.60i)T^{2} \) |
| 23 | \( 1 + (1.59 - 4.07i)T + (-16.8 - 15.6i)T^{2} \) |
| 29 | \( 1 + (0.00236 - 0.0315i)T + (-28.6 - 4.32i)T^{2} \) |
| 31 | \( 1 + (-2.33 - 1.59i)T + (11.3 + 28.8i)T^{2} \) |
| 37 | \( 1 + (0.00465 + 0.00806i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.49 + 3.61i)T + (25.5 - 32.0i)T^{2} \) |
| 47 | \( 1 + (-3.50 + 4.39i)T + (-10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (5.67 - 5.26i)T + (3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (0.483 + 2.11i)T + (-53.1 + 25.5i)T^{2} \) |
| 61 | \( 1 + (10.0 - 6.84i)T + (22.2 - 56.7i)T^{2} \) |
| 67 | \( 1 + (7.65 + 1.15i)T + (64.0 + 19.7i)T^{2} \) |
| 71 | \( 1 + (-4.78 - 12.2i)T + (-52.0 + 48.2i)T^{2} \) |
| 73 | \( 1 + (3.48 + 3.23i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (6.00 - 10.3i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.321 + 4.29i)T + (-82.0 + 12.3i)T^{2} \) |
| 89 | \( 1 + (-0.574 - 7.67i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 + (-1.00 - 1.25i)T + (-21.5 + 94.5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.78499035767620886355310955327, −10.10672237607810422539649836987, −9.182002767270719505468592345512, −8.782127954242266065256803225350, −7.73359898053578563069842070357, −6.03464638823664357241595152601, −5.54760459334831554421102709603, −4.37459244990406169952938888055, −3.59416614328172676130911056539, −2.57607067151915584622375401380,
0.45106841315564127807709458256, 1.88886447140554584112245844126, 2.91520297934393886254651266408, 4.39116330407179553720170384127, 5.98147984846942069016657645946, 6.44085261935452012622130487698, 7.55252735498001924342349792431, 7.88634979184480058159985156685, 8.859432318323596892073783509865, 10.08577225387991218573967621087
