| L(s) = 1 | + 8i·2-s − 64·4-s + 1.01e3i·7-s − 512i·8-s − 1.09e3·11-s − 1.38e3i·13-s − 8.12e3·14-s + 4.09e3·16-s − 1.47e4i·17-s + 3.99e4·19-s − 8.73e3i·22-s + 6.87e4i·23-s + 1.10e4·26-s − 6.50e4i·28-s − 1.02e5·29-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 1.11i·7-s − 0.353i·8-s − 0.247·11-s − 0.174i·13-s − 0.791·14-s + 0.250·16-s − 0.725i·17-s + 1.33·19-s − 0.174i·22-s + 1.17i·23-s + 0.123·26-s − 0.559i·28-s − 0.780·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.698289379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.698289379\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 1.01e3iT - 8.23e5T^{2} \) |
| 11 | \( 1 + 1.09e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.38e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 1.47e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 - 3.99e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 6.87e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 + 1.02e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.27e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.60e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 1.08e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 6.30e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 4.72e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + 1.49e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 2.64e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 8.27e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.26e5iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 1.41e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 9.80e5iT - 1.10e13T^{2} \) |
| 79 | \( 1 - 3.56e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.67e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 1.19e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 8.68e6iT - 8.07e13T^{2} \) |
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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
−9.922538744462511311897321470700, −9.396265159571580656491214089407, −8.392580067793038442942376623981, −7.61687115445874082041347376856, −6.61645827973704068354606505111, −5.51929515510046628631866811924, −5.05589652801223920941944494106, −3.53925079980743067108391165540, −2.47914542076382502569786738107, −1.00529851249938700541985856584,
0.40823009652290547206556487986, 1.27859499811298669053114190722, 2.56072269175267795648787656162, 3.70319121108105951203825417097, 4.47929138321471184749118462970, 5.63757886378326816180014053038, 6.88757342923967390792448964014, 7.77919914404238661338352490871, 8.738826787693138206570904406023, 9.829940955075385515334624409049
