| L(s) = 1 | − 3.20i·3-s − 11.1·5-s − 30.6i·7-s + 70.7·9-s + 168. i·11-s − 3.15·13-s + 35.8i·15-s − 229.·17-s − 151. i·19-s − 98.1·21-s + 916. i·23-s + 125.·25-s − 486. i·27-s + 1.00e3·29-s − 1.72e3i·31-s + ⋯ |
| L(s) = 1 | − 0.356i·3-s − 0.447·5-s − 0.624i·7-s + 0.873·9-s + 1.38i·11-s − 0.0186·13-s + 0.159i·15-s − 0.793·17-s − 0.418i·19-s − 0.222·21-s + 1.73i·23-s + 0.200·25-s − 0.667i·27-s + 1.19·29-s − 1.79i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.862731830\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.862731830\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 11.1T \) |
| good | 3 | \( 1 + 3.20iT - 81T^{2} \) |
| 7 | \( 1 + 30.6iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 168. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 3.15T + 2.85e4T^{2} \) |
| 17 | \( 1 + 229.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 151. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 916. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.00e3T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.72e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 841.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.52e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 1.65e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 2.66e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 3.14e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 3.26e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 767.T + 1.38e7T^{2} \) |
| 67 | \( 1 + 1.85e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 2.77e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 2.25e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 906. iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 3.83e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 1.82e3T + 6.27e7T^{2} \) |
| 97 | \( 1 - 1.58e4T + 8.85e7T^{2} \) |
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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
−11.07062588499140805378723630592, −9.979654298094822413246967139920, −9.310026976696756312135003495653, −7.72331866754223919353794498118, −7.37185565982019665252549363980, −6.33230224687519158017315311025, −4.68630713158092330146780400501, −4.01968922728505657793148483800, −2.28115971450730132366839788974, −0.939494662006656169191555794961,
0.77432998324574236116475130261, 2.57039762126910952297503228830, 3.84147450086316596507840107956, 4.86519243855579944615128121466, 6.10278662311552409629877129310, 7.07875386498053395841421639023, 8.473289519659983919449254989633, 8.855012127951690679728907670204, 10.28076839367572691853933701045, 10.84029237059211379976006460065
