| L(s) = 1 | + (−1.26 + 0.642i)2-s + (−3.62 + 0.574i)3-s + (1.17 − 1.61i)4-s + (0.654 − 4.95i)5-s + (4.20 − 3.05i)6-s + (−8.92 − 8.92i)7-s + (−0.442 + 2.79i)8-s + (4.26 − 1.38i)9-s + (2.35 + 6.66i)10-s + (−2.20 + 6.80i)11-s + (−3.33 + 6.54i)12-s + (3.81 + 1.94i)13-s + (16.9 + 5.51i)14-s + (0.472 + 18.3i)15-s + (−1.23 − 3.80i)16-s + (−1.27 − 0.202i)17-s + ⋯ |
| L(s) = 1 | + (−0.630 + 0.321i)2-s + (−1.20 + 0.191i)3-s + (0.293 − 0.404i)4-s + (0.130 − 0.991i)5-s + (0.700 − 0.508i)6-s + (−1.27 − 1.27i)7-s + (−0.0553 + 0.349i)8-s + (0.474 − 0.154i)9-s + (0.235 + 0.666i)10-s + (−0.200 + 0.618i)11-s + (−0.277 + 0.545i)12-s + (0.293 + 0.149i)13-s + (1.21 + 0.394i)14-s + (0.0315 + 1.22i)15-s + (−0.0772 − 0.237i)16-s + (−0.0750 − 0.0118i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.464 + 0.885i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.464 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.159945 - 0.264356i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.159945 - 0.264356i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.26 - 0.642i)T \) |
| 5 | \( 1 + (-0.654 + 4.95i)T \) |
| good | 3 | \( 1 + (3.62 - 0.574i)T + (8.55 - 2.78i)T^{2} \) |
| 7 | \( 1 + (8.92 + 8.92i)T + 49iT^{2} \) |
| 11 | \( 1 + (2.20 - 6.80i)T + (-97.8 - 71.1i)T^{2} \) |
| 13 | \( 1 + (-3.81 - 1.94i)T + (99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (1.27 + 0.202i)T + (274. + 89.3i)T^{2} \) |
| 19 | \( 1 + (19.5 + 26.9i)T + (-111. + 343. i)T^{2} \) |
| 23 | \( 1 + (-13.3 - 26.2i)T + (-310. + 427. i)T^{2} \) |
| 29 | \( 1 + (-14.3 + 19.7i)T + (-259. - 799. i)T^{2} \) |
| 31 | \( 1 + (0.000688 - 0.000500i)T + (296. - 913. i)T^{2} \) |
| 37 | \( 1 + (-18.6 + 36.5i)T + (-804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (6.26 + 19.2i)T + (-1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 + (-24.8 + 24.8i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-2.09 - 13.2i)T + (-2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (52.5 - 8.32i)T + (2.67e3 - 868. i)T^{2} \) |
| 59 | \( 1 + (-38.0 + 12.3i)T + (2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-5.57 + 17.1i)T + (-3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + (105. + 16.6i)T + (4.26e3 + 1.38e3i)T^{2} \) |
| 71 | \( 1 + (17.9 + 13.0i)T + (1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (2.79 + 5.48i)T + (-3.13e3 + 4.31e3i)T^{2} \) |
| 79 | \( 1 + (25.0 - 34.4i)T + (-1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-22.3 + 140. i)T + (-6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + (65.0 + 21.1i)T + (6.40e3 + 4.65e3i)T^{2} \) |
| 97 | \( 1 + (0.933 + 5.89i)T + (-8.94e3 + 2.90e3i)T^{2} \) |
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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
−15.62403826378226355256552631085, −13.55422740762027126076792889692, −12.64587075787148773443009677433, −11.19378233425716732565008270490, −10.17195931461080110307573512900, −9.116291000351141113224242881716, −7.24030303568512793225653993035, −6.08281336112480662311717068758, −4.54472395273647243007071880394, −0.42550433340169952279601394266,
2.94261650640404600366123100027, 5.96147912066002142353224639903, 6.54286606749726265729508366720, 8.565545754304300501586561600274, 10.06953711161631579133557743814, 10.95811962043971970721371642977, 12.06270036896592351975487974442, 12.91229630957724259369975368781, 14.79580265120910048680509278604, 16.05395051337862483486404710596
