L(s) = 1 | + (8.24 − 7.74i)2-s − 27i·3-s + (7.88 − 127. i)4-s − 76.0i·5-s + (−209. − 222. i)6-s − 222.·7-s + (−925. − 1.11e3i)8-s − 729·9-s + (−589. − 627. i)10-s − 1.90e3i·11-s + (−3.44e3 − 212. i)12-s + 309. i·13-s + (−1.83e3 + 1.72e3i)14-s − 2.05e3·15-s + (−1.62e4 − 2.01e3i)16-s + 1.67e4·17-s + ⋯ |
L(s) = 1 | + (0.728 − 0.684i)2-s − 0.577i·3-s + (0.0615 − 0.998i)4-s − 0.272i·5-s + (−0.395 − 0.420i)6-s − 0.245·7-s + (−0.638 − 0.769i)8-s − 0.333·9-s + (−0.186 − 0.198i)10-s − 0.431i·11-s + (−0.576 − 0.0355i)12-s + 0.0391i·13-s + (−0.178 + 0.168i)14-s − 0.157·15-s + (−0.992 − 0.122i)16-s + 0.826·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.769 + 0.638i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.732360 - 2.02843i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.732360 - 2.02843i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-8.24 + 7.74i)T \) |
| 3 | \( 1 + 27iT \) |
good | 5 | \( 1 + 76.0iT - 7.81e4T^{2} \) |
| 7 | \( 1 + 222.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.90e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 - 309. iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 1.67e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.70e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 - 7.73e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.56e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 - 2.65e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.13e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 6.94e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 9.00e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 7.71e4T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.89e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 7.04e5iT - 2.48e12T^{2} \) |
| 61 | \( 1 - 1.42e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 + 1.87e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 3.31e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.90e4T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.43e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.00e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 2.38e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.31e7T + 8.07e13T^{2} \) |
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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
−15.35606080850383664252390644179, −14.01745885278633411378496809195, −13.00262248037900166477402000082, −11.95873605428012612020039669138, −10.62219293049180299787938641197, −8.933657906152764671209435215789, −6.78173547852589269572097773862, −5.13318978617244759995592732136, −3.02827804385519465888236066080, −0.983594728582720254391568536215,
3.20152666435461835305238034415, 4.88492362005164650725829898562, 6.50887188387878640469354715852, 8.152344378713790839596574461447, 9.906826576083317500163436607658, 11.62465402207951116844687678563, 12.97370565622669055500744623183, 14.38242803872315937288716527160, 15.24274631313763369764915410348, 16.41799030271473782775488860925