| L(s) = 1 | + 3.64·3-s + 2.35i·5-s + 2.64i·7-s + 4.29·9-s + 7.29·11-s + 9.64i·13-s + 8.58i·15-s + 8.58·17-s + 15.6·19-s + 9.64i·21-s + 23.1i·23-s + 19.4·25-s − 17.1·27-s + 0.457i·29-s + 25.1i·31-s + ⋯ |
| L(s) = 1 | + 1.21·3-s + 0.470i·5-s + 0.377i·7-s + 0.476·9-s + 0.662·11-s + 0.741i·13-s + 0.572i·15-s + 0.504·17-s + 0.823·19-s + 0.459i·21-s + 1.00i·23-s + 0.778·25-s − 0.635·27-s + 0.0157i·29-s + 0.811i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.649247276\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.649247276\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
| good | 3 | \( 1 - 3.64T + 9T^{2} \) |
| 5 | \( 1 - 2.35iT - 25T^{2} \) |
| 11 | \( 1 - 7.29T + 121T^{2} \) |
| 13 | \( 1 - 9.64iT - 169T^{2} \) |
| 17 | \( 1 - 8.58T + 289T^{2} \) |
| 19 | \( 1 - 15.6T + 361T^{2} \) |
| 23 | \( 1 - 23.1iT - 529T^{2} \) |
| 29 | \( 1 - 0.457iT - 841T^{2} \) |
| 31 | \( 1 - 25.1iT - 961T^{2} \) |
| 37 | \( 1 + 24.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 54.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + 64.7T + 1.84e3T^{2} \) |
| 47 | \( 1 - 10.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 96.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 92.8T + 3.48e3T^{2} \) |
| 61 | \( 1 + 56.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 97.8T + 4.48e3T^{2} \) |
| 71 | \( 1 + 4.25iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 5.74T + 5.32e3T^{2} \) |
| 79 | \( 1 - 41.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 33.7T + 6.88e3T^{2} \) |
| 89 | \( 1 - 136.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 88.9T + 9.40e3T^{2} \) |
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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
−11.06450482726741678366824959196, −9.733999390299820052009397258578, −9.243693124609084131508073090463, −8.352121644850342106812628788896, −7.43324918432528814058259884014, −6.52071015527029220339563570225, −5.22175561739864747935753050691, −3.74957333041101778872130636963, −2.97674602293848155644516752675, −1.69361442900046782229296662599,
1.08116561485925817411176925081, 2.68819951489035490784775582452, 3.65587171872254593649891519652, 4.80358774647269554921309715102, 6.09925559900974846142377347809, 7.38662774795104776603755160911, 8.120766900837143540340674102340, 8.930073712011680702720134051935, 9.657324744006653273702939273118, 10.60556823456809859841664569338
