| L(s) = 1 | + (−1.23 − 3.80i)2-s − 29.2·3-s + (−12.9 + 9.40i)4-s + (72.0 − 52.3i)5-s + (36.2 + 111. i)6-s + (−58.7 + 180. i)7-s + (51.7 + 37.6i)8-s + 615.·9-s + (−288. − 209. i)10-s + (102. + 74.1i)11-s + (379. − 275. i)12-s + (9.10 + 28.0i)13-s + 760.·14-s + (−2.11e3 + 1.53e3i)15-s + (79.1 − 243. i)16-s + (−1.17e3 − 853. i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.672i)2-s − 1.87·3-s + (−0.404 + 0.293i)4-s + (1.28 − 0.936i)5-s + (0.410 + 1.26i)6-s + (−0.452 + 1.39i)7-s + (0.286 + 0.207i)8-s + 2.53·9-s + (−0.911 − 0.662i)10-s + (0.254 + 0.184i)11-s + (0.760 − 0.552i)12-s + (0.0149 + 0.0459i)13-s + 1.03·14-s + (−2.42 + 1.76i)15-s + (0.0772 − 0.237i)16-s + (−0.986 − 0.716i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.424 + 0.905i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.424 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.415253 - 0.653579i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.415253 - 0.653579i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 + 3.80i)T \) |
| 41 | \( 1 + (-1.04e4 - 2.38e3i)T \) |
| good | 3 | \( 1 + 29.2T + 243T^{2} \) |
| 5 | \( 1 + (-72.0 + 52.3i)T + (965. - 2.97e3i)T^{2} \) |
| 7 | \( 1 + (58.7 - 180. i)T + (-1.35e4 - 9.87e3i)T^{2} \) |
| 11 | \( 1 + (-102. - 74.1i)T + (4.97e4 + 1.53e5i)T^{2} \) |
| 13 | \( 1 + (-9.10 - 28.0i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (1.17e3 + 853. i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-806. + 2.48e3i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + (-708. - 2.18e3i)T + (-5.20e6 + 3.78e6i)T^{2} \) |
| 29 | \( 1 + (1.63e3 - 1.18e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (424. + 308. i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-7.99e3 + 5.81e3i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 43 | \( 1 + (2.43e3 + 7.48e3i)T + (-1.18e8 + 8.64e7i)T^{2} \) |
| 47 | \( 1 + (7.70e3 + 2.37e4i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-2.52e3 + 1.83e3i)T + (1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (9.97e3 + 3.07e4i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-1.59e4 + 4.89e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-1.66e4 + 1.20e4i)T + (4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (-4.82e3 - 3.50e3i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + 4.27e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.83e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.17e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-5.94e3 + 1.83e4i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (1.00e5 - 7.29e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
| 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
−12.79266947511093812706491596690, −11.83646819910288432988929742638, −11.07497685895217957762885981438, −9.546999462285047395752703066534, −9.226295272212950974239291237210, −6.69532688399840639440964310254, −5.53287931665530488308144740157, −4.89953407551903196155336616703, −2.04512791105038234552459163245, −0.52844210565309645783709832229,
1.17071922920450298699907832338, 4.27914785121309188553179971665, 5.93888498809196989230059565383, 6.36945558374409551770774227984, 7.34231219610735691585281346253, 9.790648329068205315230874358920, 10.38987408716955898406334221799, 11.10800326041969928522659996623, 12.78318571183637309724922866090, 13.64527821986103456201321497669
