| L(s) = 1 | + 9.07i·5-s − 2.64·7-s − 2.06i·11-s + 4.46·13-s − 9.11i·17-s − 5.96·19-s − 19.7i·23-s − 57.2·25-s + 6.15i·29-s + 53.8·31-s − 23.9i·35-s + 36.4·37-s − 64.4i·41-s + 73.4·43-s + 22.5i·47-s + ⋯ |
| L(s) = 1 | + 1.81i·5-s − 0.377·7-s − 0.188i·11-s + 0.343·13-s − 0.536i·17-s − 0.314·19-s − 0.856i·23-s − 2.29·25-s + 0.212i·29-s + 1.73·31-s − 0.685i·35-s + 0.984·37-s − 1.57i·41-s + 1.70·43-s + 0.478i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.038900691\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.038900691\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
| good | 5 | \( 1 - 9.07iT - 25T^{2} \) |
| 11 | \( 1 + 2.06iT - 121T^{2} \) |
| 13 | \( 1 - 4.46T + 169T^{2} \) |
| 17 | \( 1 + 9.11iT - 289T^{2} \) |
| 19 | \( 1 + 5.96T + 361T^{2} \) |
| 23 | \( 1 + 19.7iT - 529T^{2} \) |
| 29 | \( 1 - 6.15iT - 841T^{2} \) |
| 31 | \( 1 - 53.8T + 961T^{2} \) |
| 37 | \( 1 - 36.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 64.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 73.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 22.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 14.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 71.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 38.8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 92.4T + 4.48e3T^{2} \) |
| 71 | \( 1 + 29.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 62.3T + 5.32e3T^{2} \) |
| 79 | \( 1 - 120.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 110. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 125. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 89.6T + 9.40e3T^{2} \) |
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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
−8.270193292704115986552481034196, −7.50856372473674578110028957860, −6.84336713415677760326375451613, −6.31685020768325240372612456953, −5.69359493969924968763975826258, −4.43011886010943057608462740679, −3.66049247462478956773381528515, −2.77368245741360816194314018300, −2.34726811456185914884873027822, −0.69418418008487042953940652778,
0.67147657396298594262272893742, 1.38828399520585577843519404457, 2.50646019379601063650489974529, 3.77854600984269754270244384148, 4.40046535021392164142390983984, 5.10754798727250969400927497509, 5.90147347704697544776561215032, 6.51491657524021897754009173247, 7.72684719226660903807118612857, 8.209970054439159080072182406224
