| L(s) = 1 | + (−1.37 − 4.24i)2-s + (6.03 + 4.38i)3-s + (−9.65 + 7.01i)4-s + (−2.61 + 8.04i)5-s + (10.2 − 31.6i)6-s + (−16.3 + 11.9i)7-s + (14.1 + 10.3i)8-s + (8.87 + 27.3i)9-s + 37.7·10-s − 89.0·12-s + (19.0 + 58.5i)13-s + (73.1 + 53.1i)14-s + (−51.1 + 37.1i)15-s + (−5.29 + 16.3i)16-s + (−21.4 + 65.9i)17-s + (103. − 75.3i)18-s + ⋯ |
| L(s) = 1 | + (−0.487 − 1.50i)2-s + (1.16 + 0.844i)3-s + (−1.20 + 0.876i)4-s + (−0.233 + 0.719i)5-s + (0.700 − 2.15i)6-s + (−0.884 + 0.642i)7-s + (0.626 + 0.455i)8-s + (0.328 + 1.01i)9-s + 1.19·10-s − 2.14·12-s + (0.405 + 1.24i)13-s + (1.39 + 1.01i)14-s + (−0.879 + 0.639i)15-s + (−0.0827 + 0.254i)16-s + (−0.305 + 0.941i)17-s + (1.35 − 0.986i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.780 - 0.625i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.18216 + 0.415429i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.18216 + 0.415429i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| good | 2 | \( 1 + (1.37 + 4.24i)T + (-6.47 + 4.70i)T^{2} \) |
| 3 | \( 1 + (-6.03 - 4.38i)T + (8.34 + 25.6i)T^{2} \) |
| 5 | \( 1 + (2.61 - 8.04i)T + (-101. - 73.4i)T^{2} \) |
| 7 | \( 1 + (16.3 - 11.9i)T + (105. - 326. i)T^{2} \) |
| 13 | \( 1 + (-19.0 - 58.5i)T + (-1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (21.4 - 65.9i)T + (-3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (5.80 + 4.21i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 - 50.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-115. + 84.1i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (78.7 + 242. i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (272. - 197. i)T + (1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (144. + 104. i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 + 55.7T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-207. - 150. i)T + (3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-65.9 - 203. i)T + (-1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-645. + 468. i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-52.1 + 160. i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 + 366.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (241. - 743. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-774. + 562. i)T + (1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-180. - 556. i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (202. - 624. i)T + (-4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 - 72.6T + 7.04e5T^{2} \) |
| 97 | \( 1 + (302. + 931. i)T + (-7.38e5 + 5.36e5i)T^{2} \) |
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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
−12.95317970989242380852870911800, −11.80546397726331342283085656041, −10.82705905160071365512318493177, −9.906729058526152094221050144643, −9.162229535949142285945147199930, −8.451106257882439960868144908329, −6.54994305308216649839059789085, −4.08530943963301903054935621307, −3.25101156527675485447184684161, −2.22260230430884443377663432499,
0.67827274202671059337647605213, 3.18416487564280082301433743386, 5.18926603401687552959204944172, 6.76524875102902750739084272914, 7.37634937985421845372444672366, 8.497361878728584997488416569741, 8.928331328783762289190280234952, 10.29509436568366819635041098333, 12.38473456604276238677292637693, 13.26278375755228588180740044257
