L(s) = 1 | − 6.89i·3-s + (−10.7 + 2.89i)5-s − 12.6i·7-s − 20.5·9-s − 59.1·11-s + 42.2i·13-s + (20 + 74.4i)15-s − 126. i·17-s − 19.1·19-s − 87.5·21-s − 78.3i·23-s + (108. − 62.6i)25-s − 44.1i·27-s + 148.·29-s + 139.·31-s + ⋯ |
L(s) = 1 | − 1.32i·3-s + (−0.965 + 0.259i)5-s − 0.685i·7-s − 0.762·9-s − 1.62·11-s + 0.900i·13-s + (0.344 + 1.28i)15-s − 1.80i·17-s − 0.231·19-s − 0.910·21-s − 0.709i·23-s + (0.865 − 0.500i)25-s − 0.314i·27-s + 0.950·29-s + 0.806·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.259i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.100525 - 0.762124i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.100525 - 0.762124i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (10.7 - 2.89i)T \) |
good | 3 | \( 1 + 6.89iT - 27T^{2} \) |
| 7 | \( 1 + 12.6iT - 343T^{2} \) |
| 11 | \( 1 + 59.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 42.2iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 126. iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 19.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 78.3iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 148.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 139.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 66.5iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 203.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 288. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 360. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 686. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 83.1T + 2.05e5T^{2} \) |
| 61 | \( 1 + 208.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 192. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 500.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 122. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 289.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 573. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 565.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 643. iT - 9.12e5T^{2} \) |
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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
−13.41733050393229321136191021734, −12.32028534985584654729697258728, −11.46911320426340671278095720569, −10.25527865668363558814914410982, −8.370126726547041795488768742635, −7.42798408139642627157583569935, −6.75096687759686485093148992146, −4.70190385875995423008554402988, −2.67907010140260838678315132009, −0.46927494006564300260224132921,
3.16448488694198354765990803270, 4.54568818729749088897292605765, 5.65735560745928232421015020108, 7.87776854377764400457506383481, 8.688003410234071073683043577782, 10.21541214587252019342893328122, 10.78542590624163208789060094282, 12.19145247475663599124636096891, 13.14761173250572637682422919201, 15.05039349881851024422042397753