| L(s) = 1 | − 9.35i·3-s + 21.0·5-s + 9.13·7-s − 6.44·9-s − 229.·11-s + 31.9i·13-s − 196. i·15-s − 200.·17-s + (−351. − 84.1i)19-s − 85.4i·21-s + 317.·23-s − 183.·25-s − 697. i·27-s − 222. i·29-s − 1.61e3i·31-s + ⋯ |
| L(s) = 1 | − 1.03i·3-s + 0.840·5-s + 0.186·7-s − 0.0795·9-s − 1.89·11-s + 0.189i·13-s − 0.873i·15-s − 0.694·17-s + (−0.972 − 0.233i)19-s − 0.193i·21-s + 0.600·23-s − 0.293·25-s − 0.956i·27-s − 0.264i·29-s − 1.68i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.233i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.972 - 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.7371552964\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7371552964\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (351. + 84.1i)T \) |
| good | 3 | \( 1 + 9.35iT - 81T^{2} \) |
| 5 | \( 1 - 21.0T + 625T^{2} \) |
| 7 | \( 1 - 9.13T + 2.40e3T^{2} \) |
| 11 | \( 1 + 229.T + 1.46e4T^{2} \) |
| 13 | \( 1 - 31.9iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 200.T + 8.35e4T^{2} \) |
| 23 | \( 1 - 317.T + 2.79e5T^{2} \) |
| 29 | \( 1 + 222. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.61e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.60e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 1.95e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.40e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 2.21e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 1.73e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 3.70e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 44.0T + 1.38e7T^{2} \) |
| 67 | \( 1 - 3.06e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 5.45e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 729.T + 2.83e7T^{2} \) |
| 79 | \( 1 - 9.50e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 1.48e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 3.85e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.13e4iT - 8.85e7T^{2} \) |
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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
−10.55579538611189476749545336159, −9.737557250848943781652646527436, −8.444566354832639663330245330727, −7.65922834116752869133987961633, −6.65714676676074738813242739703, −5.73501280995927419964338537238, −4.59010655192951510517703908747, −2.59453165510263027547937515359, −1.83343488951466989060783870713, −0.20059057320650255138891229126,
1.95921742380903416429946539258, 3.23375431465573152784793478335, 4.74067245878222917844025605704, 5.29633892690530605519771664019, 6.58140637092674961797589062834, 7.922965145743790929847730115747, 8.909963701278814897946640497432, 9.940737259546625712774152301222, 10.48096378484919047797834318227, 11.15342406869346788962413944477
