| L(s) = 1 | + (−0.784 − 1.35i)3-s + (−24.5 − 42.3i)7-s + (39.2 − 68.0i)9-s + (−11.7 − 20.3i)11-s + 136.·13-s + (131. + 227. i)17-s + (387. + 223. i)19-s + (−38.3 + 66.6i)21-s + (648. + 374. i)23-s − 250.·27-s − 406.·29-s + (−584. + 337. i)31-s + (−18.4 + 31.9i)33-s + (645. + 372. i)37-s + (−106. − 185. i)39-s + ⋯ |
| L(s) = 1 | + (−0.0871 − 0.151i)3-s + (−0.501 − 0.865i)7-s + (0.484 − 0.839i)9-s + (−0.0971 − 0.168i)11-s + 0.806·13-s + (0.454 + 0.786i)17-s + (1.07 + 0.619i)19-s + (−0.0869 + 0.151i)21-s + (1.22 + 0.708i)23-s − 0.343·27-s − 0.483·29-s + (−0.607 + 0.350i)31-s + (−0.0169 + 0.0293i)33-s + (0.471 + 0.272i)37-s + (−0.0702 − 0.121i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.112261209\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.112261209\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (24.5 + 42.3i)T \) |
| good | 3 | \( 1 + (0.784 + 1.35i)T + (-40.5 + 70.1i)T^{2} \) |
| 11 | \( 1 + (11.7 + 20.3i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 - 136.T + 2.85e4T^{2} \) |
| 17 | \( 1 + (-131. - 227. i)T + (-4.17e4 + 7.23e4i)T^{2} \) |
| 19 | \( 1 + (-387. - 223. i)T + (6.51e4 + 1.12e5i)T^{2} \) |
| 23 | \( 1 + (-648. - 374. i)T + (1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + 406.T + 7.07e5T^{2} \) |
| 31 | \( 1 + (584. - 337. i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + (-645. - 372. i)T + (9.37e5 + 1.62e6i)T^{2} \) |
| 41 | \( 1 + 2.47e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 2.63e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (334. - 579. i)T + (-2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 + (1.75e3 - 1.01e3i)T + (3.94e6 - 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-1.01e3 + 585. i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (2.02e3 + 1.16e3i)T + (6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-6.54e3 + 3.77e3i)T + (1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 - 7.57e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (-1.89e3 - 3.28e3i)T + (-1.41e7 + 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-3.89e3 + 6.75e3i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 - 2.18e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + (8.05e3 + 4.64e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 - 9.98e3T + 8.85e7T^{2} \) |
| 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
−9.689347806033623953510820598783, −8.981220274709314256636672436479, −7.78704622309247144140991889484, −7.07212974545706711247335417261, −6.22022921161667057831606038491, −5.27968105974971188673304065730, −3.74843669179406048002676374853, −3.45672876644226539948977370072, −1.51981169596749391062039551301, −0.64188659835651935142996673318,
1.00010464750136170943107109039, 2.41767808830725214311194952972, 3.34485570072541265723997828397, 4.75040963051655304827739766052, 5.41918981130907074703464210600, 6.49322537729246587832109328994, 7.42290920277910531947998175252, 8.322970125763891306364428778031, 9.375388265137903528482307820071, 9.803539455365964199905802129884
