| L(s) = 1 | + (−1.58 − 1.18i)2-s + (2.15 − 2.15i)3-s + (0.535 + 1.82i)4-s + (−2.10 − 0.762i)5-s + (−5.94 + 0.855i)6-s + (−2.51 + 0.546i)7-s + (−0.0688 + 0.184i)8-s − 6.25i·9-s + (2.42 + 3.69i)10-s + (−2.00 + 2.64i)11-s + (5.07 + 2.77i)12-s + (−4.23 − 2.31i)13-s + (4.61 + 2.10i)14-s + (−6.16 + 2.88i)15-s + (3.52 − 2.26i)16-s + (5.73 + 0.409i)17-s + ⋯ |
| L(s) = 1 | + (−1.11 − 0.836i)2-s + (1.24 − 1.24i)3-s + (0.267 + 0.911i)4-s + (−0.940 − 0.341i)5-s + (−2.42 + 0.349i)6-s + (−0.949 + 0.206i)7-s + (−0.0243 + 0.0653i)8-s − 2.08i·9-s + (0.765 + 1.16i)10-s + (−0.603 + 0.797i)11-s + (1.46 + 0.799i)12-s + (−1.17 − 0.642i)13-s + (1.23 + 0.563i)14-s + (−1.59 + 0.744i)15-s + (0.881 − 0.566i)16-s + (1.39 + 0.0994i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.180 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.180 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.155438 + 0.129501i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.155438 + 0.129501i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.10 + 0.762i)T \) |
| 11 | \( 1 + (2.00 - 2.64i)T \) |
| good | 2 | \( 1 + (1.58 + 1.18i)T + (0.563 + 1.91i)T^{2} \) |
| 3 | \( 1 + (-2.15 + 2.15i)T - 3iT^{2} \) |
| 7 | \( 1 + (2.51 - 0.546i)T + (6.36 - 2.90i)T^{2} \) |
| 13 | \( 1 + (4.23 + 2.31i)T + (7.02 + 10.9i)T^{2} \) |
| 17 | \( 1 + (-5.73 - 0.409i)T + (16.8 + 2.41i)T^{2} \) |
| 19 | \( 1 + (0.804 - 0.928i)T + (-2.70 - 18.8i)T^{2} \) |
| 23 | \( 1 + (0.149 - 0.685i)T + (-20.9 - 9.55i)T^{2} \) |
| 29 | \( 1 + (5.08 - 5.86i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (-1.71 - 0.502i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (2.48 + 4.55i)T + (-20.0 + 31.1i)T^{2} \) |
| 41 | \( 1 + (8.36 - 1.20i)T + (39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-3.10 + 8.32i)T + (-32.4 - 28.1i)T^{2} \) |
| 47 | \( 1 + (-6.47 + 4.84i)T + (13.2 - 45.0i)T^{2} \) |
| 53 | \( 1 + (9.80 - 2.13i)T + (48.2 - 22.0i)T^{2} \) |
| 59 | \( 1 + (12.6 + 1.82i)T + (56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (1.50 + 0.215i)T + (58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-4.26 + 5.69i)T + (-18.8 - 64.2i)T^{2} \) |
| 71 | \( 1 + (6.25 - 7.21i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (4.89 + 1.06i)T + (66.4 + 30.3i)T^{2} \) |
| 79 | \( 1 + (1.63 - 3.57i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-2.58 - 11.9i)T + (-75.4 + 34.4i)T^{2} \) |
| 89 | \( 1 + (-2.90 + 2.51i)T + (12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (1.92 - 5.15i)T + (-73.3 - 63.5i)T^{2} \) |
| 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
−9.746766478602620755644630615738, −9.109804261561958506149382957614, −8.224995583991218393541008519202, −7.60311278379414978394371239381, −7.10586939866812651057241856818, −5.38700890187100416475582394057, −3.44911874807636198054956510622, −2.80190958444689286978129355438, −1.63692666818386784140902694553, −0.13481008264198381502010660006,
2.91902639799364463939983202036, 3.55585063771977727515347387176, 4.70445728182001804345674905321, 6.21841062947857893811887908015, 7.48679252997938027352193121738, 7.83776610841487672761867751980, 8.715182870460892795759895283407, 9.539948840000941929320262416428, 9.990397908036152409683281918948, 10.73689842139099756859782370857
