| L(s) = 1 | − 1.73i·3-s − 4·5-s − 6.92i·7-s − 2.99·9-s + 6.92i·11-s + 6.92i·15-s − 18·17-s + 20.7i·19-s − 11.9·21-s − 41.5i·23-s − 9·25-s + 5.19i·27-s + 4·29-s + 48.4i·31-s + 11.9·33-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.800·5-s − 0.989i·7-s − 0.333·9-s + 0.629i·11-s + 0.461i·15-s − 1.05·17-s + 1.09i·19-s − 0.571·21-s − 1.80i·23-s − 0.359·25-s + 0.192i·27-s + 0.137·29-s + 1.56i·31-s + 0.363·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5233423087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5233423087\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| good | 5 | \( 1 + 4T + 25T^{2} \) |
| 7 | \( 1 + 6.92iT - 49T^{2} \) |
| 11 | \( 1 - 6.92iT - 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 + 18T + 289T^{2} \) |
| 19 | \( 1 - 20.7iT - 361T^{2} \) |
| 23 | \( 1 + 41.5iT - 529T^{2} \) |
| 29 | \( 1 - 4T + 841T^{2} \) |
| 31 | \( 1 - 48.4iT - 961T^{2} \) |
| 37 | \( 1 - 72T + 1.36e3T^{2} \) |
| 41 | \( 1 + 18T + 1.68e3T^{2} \) |
| 43 | \( 1 - 62.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 41.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 44T + 2.80e3T^{2} \) |
| 59 | \( 1 - 62.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 72T + 3.72e3T^{2} \) |
| 67 | \( 1 - 20.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 41.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 82T + 5.32e3T^{2} \) |
| 79 | \( 1 + 62.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 131. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 126T + 7.92e3T^{2} \) |
| 97 | \( 1 - 110T + 9.40e3T^{2} \) |
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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
−10.51441758882441606494961169666, −9.497887904497105797759077795865, −8.329739780746190286146620614345, −7.78453576834799799880328189499, −6.91159832148606095664768353703, −6.22909918451600164528141805674, −4.65001419196546255673891595530, −4.05663567865512245022222782329, −2.69530604468412537482281238030, −1.22954929847052952831888931752,
0.19259348445552987089425141209, 2.28913126131634597050883117703, 3.42116533377291872565787190598, 4.37006153693629314802574784735, 5.39929534159301542481198665948, 6.23874225324380320766477949299, 7.46080457157381997766701586741, 8.303843887130542466431687944384, 9.123393359606238580007183979689, 9.683812991124919364960528717269
