| L(s) = 1 | + 2.82·2-s + 11.3·3-s + 8.00·4-s + 11.1i·5-s + 31.9·6-s + 52.5i·7-s + 22.6·8-s + 46.8·9-s + 31.6i·10-s − 103. i·11-s + 90.4·12-s + 233.·13-s + 148. i·14-s + 126. i·15-s + 64.0·16-s + 11.4i·17-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.25·3-s + 0.500·4-s + 0.447i·5-s + 0.888·6-s + 1.07i·7-s + 0.353·8-s + 0.578·9-s + 0.316i·10-s − 0.857i·11-s + 0.628·12-s + 1.38·13-s + 0.757i·14-s + 0.561i·15-s + 0.250·16-s + 0.0394i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(4.698289192\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.698289192\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2.82T \) |
| 5 | \( 1 - 11.1iT \) |
| 23 | \( 1 + (-334. - 409. i)T \) |
| good | 3 | \( 1 - 11.3T + 81T^{2} \) |
| 7 | \( 1 - 52.5iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 103. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 233.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 11.4iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 403. iT - 1.30e5T^{2} \) |
| 29 | \( 1 + 616.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.17e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 912. iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 103.T + 2.82e6T^{2} \) |
| 43 | \( 1 + 901. iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 2.60e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + 3.03e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 4.64e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + 3.50e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 3.51e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 3.13e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 3.15e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 1.85e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 1.20e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 6.78e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 6.34e3iT - 8.85e7T^{2} \) |
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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
−11.68486762556652418179267817311, −10.89533654312415206116760608787, −9.515518357086244616543393671832, −8.567647882530651205477315514500, −7.890423836578948616935100741321, −6.36169443399761149316628087445, −5.53635240848108804133024118528, −3.69495544187491634941938692052, −3.07601791760698218888790397801, −1.81403650379172649919810825411,
1.23593056104346695704071911458, 2.74279673581268228549331128983, 3.89265430944206808587540973001, 4.75507731657835926014332345637, 6.44221805489795609386873830044, 7.47100631506094392084302368306, 8.422977456439584683882726882513, 9.369119711902207365590040995027, 10.51091166155590566310898604793, 11.48812186548705301437510995193
