| L(s) = 1 | + (−11.5 − 11.0i)2-s + (11.6 + 255. i)4-s − 279.·5-s + 3.32e3i·7-s + (2.69e3 − 3.08e3i)8-s + (3.23e3 + 3.08e3i)10-s − 6.36e3i·11-s − 3.07e4·13-s + (3.67e4 − 3.84e4i)14-s + (−6.52e4 + 5.94e3i)16-s − 1.22e5·17-s − 7.55e3i·19-s + (−3.24e3 − 7.14e4i)20-s + (−7.03e4 + 7.35e4i)22-s + 4.06e5i·23-s + ⋯ |
| L(s) = 1 | + (−0.722 − 0.690i)2-s + (0.0453 + 0.998i)4-s − 0.447·5-s + 1.38i·7-s + (0.657 − 0.753i)8-s + (0.323 + 0.308i)10-s − 0.434i·11-s − 1.07·13-s + (0.956 − 1.00i)14-s + (−0.995 + 0.0906i)16-s − 1.46·17-s − 0.0579i·19-s + (−0.0202 − 0.446i)20-s + (−0.300 + 0.314i)22-s + 1.45i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0453 + 0.998i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.0453 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.4601222739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4601222739\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (11.5 + 11.0i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 279.T \) |
| good | 7 | \( 1 - 3.32e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 6.36e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 3.07e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 1.22e5T + 6.97e9T^{2} \) |
| 19 | \( 1 + 7.55e3iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 4.06e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 8.28e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + 1.39e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 + 3.86e5T + 3.51e12T^{2} \) |
| 41 | \( 1 - 9.72e5T + 7.98e12T^{2} \) |
| 43 | \( 1 - 4.61e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 1.87e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 8.01e6T + 6.22e13T^{2} \) |
| 59 | \( 1 - 6.18e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 9.79e6T + 1.91e14T^{2} \) |
| 67 | \( 1 + 1.19e6iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 3.62e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 3.35e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 1.22e6iT - 1.51e15T^{2} \) |
| 83 | \( 1 + 6.96e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 5.58e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + 5.97e7T + 7.83e15T^{2} \) |
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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
−11.24031642563184378700345846096, −9.714749910818859460582304096008, −9.071356792425537302024769577781, −8.122006894359780958417601537529, −7.09006667210236737296518422049, −5.60188380749259629923309157052, −4.17780195630913093436581685546, −2.80958879542944321931861610752, −1.94684601736557377619903700172, −0.21691210258230846929537326733,
0.65558462608241000387585440784, 2.15836582007195857433001917440, 4.12023034607712814521678969769, 5.01669592402809577627150951687, 6.77317940379807926002280168365, 7.16526371399871882357286244760, 8.257621294509262208136667609430, 9.330838858874650161290975381693, 10.42230211009087965247711343547, 10.95228153666593333776657618560
