| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.288i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.957 + 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.421663 - 2.86582i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.421663 - 2.86582i\) |
| \(L(\frac{5}{2})\) |
|
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
−12.40546365067232479575141402371, −11.91327198312928060259780645503, −10.69607235104251838799671944670, −9.140015260074177297905055946855, −8.596547067000587242772681400338, −7.27847891636597741665295262442, −5.10739376705980664657870354574, −3.82905139980090057138219568997, −2.29254836935325206343480497179, −1.47552559242952889230873408598,
3.08602206031028110432264237565, 4.33180217796015761009102346315, 5.43186535484947829828586855767, 7.24051982158270560077758270395, 7.75003476022109841244859524187, 8.811613317320396971430896559522, 10.11340763926000574658088557230, 11.17919832035434774826887936477, 13.07615437093099934114269139926, 14.06154206123776178164906405996
