L(s) = 1 | + 932. i·2-s + (−4.23e3 − 1.92e4i)3-s − 6.07e5·4-s − 1.14e6i·5-s + (1.79e7 − 3.94e6i)6-s − 9.81e6·7-s − 3.22e8i·8-s + (−3.51e8 + 1.62e8i)9-s + 1.06e9·10-s − 2.16e9i·11-s + (2.57e9 + 1.16e10i)12-s − 1.47e10·13-s − 9.15e9i·14-s + (−2.19e10 + 4.83e9i)15-s + 1.41e11·16-s − 1.36e10i·17-s + ⋯ |
L(s) = 1 | + 1.82i·2-s + (−0.215 − 0.976i)3-s − 2.31·4-s − 0.584i·5-s + (1.77 − 0.391i)6-s − 0.243·7-s − 2.40i·8-s + (−0.907 + 0.420i)9-s + 1.06·10-s − 0.917i·11-s + (0.498 + 2.26i)12-s − 1.38·13-s − 0.442i·14-s + (−0.571 + 0.125i)15-s + 2.05·16-s − 0.115i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.215 + 0.976i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{19}{2})\) |
\(\approx\) |
\(0.288210 - 0.231636i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.288210 - 0.231636i\) |
\(L(10)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (4.23e3 + 1.92e4i)T \) |
good | 2 | \( 1 - 932. iT - 2.62e5T^{2} \) |
| 5 | \( 1 + 1.14e6iT - 3.81e12T^{2} \) |
| 7 | \( 1 + 9.81e6T + 1.62e15T^{2} \) |
| 11 | \( 1 + 2.16e9iT - 5.55e18T^{2} \) |
| 13 | \( 1 + 1.47e10T + 1.12e20T^{2} \) |
| 17 | \( 1 + 1.36e10iT - 1.40e22T^{2} \) |
| 19 | \( 1 + 2.38e11T + 1.04e23T^{2} \) |
| 23 | \( 1 - 5.70e11iT - 3.24e24T^{2} \) |
| 29 | \( 1 - 1.35e13iT - 2.10e26T^{2} \) |
| 31 | \( 1 + 1.17e13T + 6.99e26T^{2} \) |
| 37 | \( 1 + 1.26e14T + 1.68e28T^{2} \) |
| 41 | \( 1 + 3.28e14iT - 1.07e29T^{2} \) |
| 43 | \( 1 - 8.40e14T + 2.52e29T^{2} \) |
| 47 | \( 1 + 9.97e14iT - 1.25e30T^{2} \) |
| 53 | \( 1 + 5.45e15iT - 1.08e31T^{2} \) |
| 59 | \( 1 - 8.03e14iT - 7.50e31T^{2} \) |
| 61 | \( 1 - 1.04e16T + 1.36e32T^{2} \) |
| 67 | \( 1 + 1.50e16T + 7.40e32T^{2} \) |
| 71 | \( 1 - 4.47e16iT - 2.10e33T^{2} \) |
| 73 | \( 1 + 3.92e16T + 3.46e33T^{2} \) |
| 79 | \( 1 + 7.20e16T + 1.43e34T^{2} \) |
| 83 | \( 1 - 1.02e17iT - 3.49e34T^{2} \) |
| 89 | \( 1 + 5.12e17iT - 1.22e35T^{2} \) |
| 97 | \( 1 + 7.28e17T + 5.77e35T^{2} \) |
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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
−22.36990056644114839035632200611, −19.14884204022160949633830844544, −17.47352518751496202128310732232, −16.39736593526527573529566357611, −14.35510480299154864627661117601, −12.80916506798079875930602272471, −8.654058992399360153589039282056, −7.04734512498114625163189500203, −5.36882103519765579444020504273, −0.21279680117724765709893767246,
2.64563251124459065798211352237, 4.47006469852609182577849627871, 9.549559944850293134810634436874, 10.69116032561887396356713062181, 12.33252076187058749702464154719, 14.66553677059359858556372116360, 17.47922127685418936653693802840, 19.32934560951804847129823707393, 20.66093401767692633877397390720, 21.99866719651942617296410302882