L(s) = 1 | + (1.25e5 − 3.63e4i)2-s + 1.93e8i·3-s + (1.45e10 − 9.14e9i)4-s − 8.66e11·5-s + (7.02e12 + 2.43e13i)6-s − 8.12e13i·7-s + (1.49e15 − 1.68e15i)8-s − 2.07e16·9-s + (−1.09e17 + 3.14e16i)10-s + 5.42e17i·11-s + (1.76e18 + 2.81e18i)12-s − 1.50e19·13-s + (−2.95e18 − 1.02e19i)14-s − 1.67e20i·15-s + (1.27e20 − 2.66e20i)16-s − 1.97e20·17-s + ⋯ |
L(s) = 1 | + (0.960 − 0.277i)2-s + 1.49i·3-s + (0.846 − 0.532i)4-s − 1.13·5-s + (0.415 + 1.43i)6-s − 0.349i·7-s + (0.665 − 0.746i)8-s − 1.24·9-s + (−1.09 + 0.314i)10-s + 1.07i·11-s + (0.797 + 1.26i)12-s − 1.73·13-s + (−0.0967 − 0.335i)14-s − 1.70i·15-s + (0.432 − 0.901i)16-s − 0.238·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.846 + 0.532i)\, \overline{\Lambda}(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & (-0.846 + 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{35}{2})\) |
\(\approx\) |
\(0.1487307861\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1487307861\) |
\(L(18)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.25e5 + 3.63e4i)T \) |
good | 3 | \( 1 - 1.93e8iT - 1.66e16T^{2} \) |
| 5 | \( 1 + 8.66e11T + 5.82e23T^{2} \) |
| 7 | \( 1 + 8.12e13iT - 5.41e28T^{2} \) |
| 11 | \( 1 - 5.42e17iT - 2.55e35T^{2} \) |
| 13 | \( 1 + 1.50e19T + 7.48e37T^{2} \) |
| 17 | \( 1 + 1.97e20T + 6.84e41T^{2} \) |
| 19 | \( 1 + 6.27e21iT - 3.00e43T^{2} \) |
| 23 | \( 1 + 2.46e23iT - 1.98e46T^{2} \) |
| 29 | \( 1 + 7.75e24T + 5.26e49T^{2} \) |
| 31 | \( 1 - 1.56e24iT - 5.08e50T^{2} \) |
| 37 | \( 1 - 9.31e25T + 2.08e53T^{2} \) |
| 41 | \( 1 + 2.42e27T + 6.83e54T^{2} \) |
| 43 | \( 1 - 8.07e27iT - 3.45e55T^{2} \) |
| 47 | \( 1 - 3.66e28iT - 7.10e56T^{2} \) |
| 53 | \( 1 + 9.36e28T + 4.22e58T^{2} \) |
| 59 | \( 1 - 1.58e30iT - 1.61e60T^{2} \) |
| 61 | \( 1 + 2.50e30T + 5.02e60T^{2} \) |
| 67 | \( 1 + 3.53e30iT - 1.22e62T^{2} \) |
| 71 | \( 1 - 2.74e31iT - 8.76e62T^{2} \) |
| 73 | \( 1 - 2.97e31T + 2.25e63T^{2} \) |
| 79 | \( 1 + 3.89e31iT - 3.30e64T^{2} \) |
| 83 | \( 1 - 1.42e32iT - 1.77e65T^{2} \) |
| 89 | \( 1 + 9.61e32T + 1.90e66T^{2} \) |
| 97 | \( 1 + 1.00e34T + 3.55e67T^{2} \) |
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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
−16.70182857145942706598745334130, −15.37214905737528943417837684394, −14.73187299231382567989644631831, −12.39637699061635443387258546461, −11.00505031545854542133313948482, −9.737516424357559195142094012688, −7.24247054190421723432495175188, −4.74299910577326723588855690389, −4.29743277219232242394727759639, −2.72433726568946652421558102877,
0.03101637715391467973773578133, 1.95397172089531446702733261190, 3.50217035057573867991593486181, 5.56795818409594452814946503448, 7.17826740121118541165522421261, 7.999048758861229607764439363594, 11.61250334733851020074686489928, 12.33434931883553527548684578691, 13.73946937311523901065578980841, 15.22859035621667448084012264306