| L(s) = 1 | + 4.89·5-s − 7.74i·7-s + 12.6i·11-s − 15.4·13-s + 25.2·17-s + 8i·19-s + 39.1i·23-s − 1.00·25-s − 24.4·29-s − 7.74i·31-s − 37.9i·35-s − 46.4·37-s − 25.2·41-s + 40i·43-s + 39.1i·47-s + ⋯ |
| L(s) = 1 | + 0.979·5-s − 1.10i·7-s + 1.14i·11-s − 1.19·13-s + 1.48·17-s + 0.421i·19-s + 1.70i·23-s − 0.0400·25-s − 0.844·29-s − 0.249i·31-s − 1.08i·35-s − 1.25·37-s − 0.617·41-s + 0.930i·43-s + 0.833i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.666405523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.666405523\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 4.89T + 25T^{2} \) |
| 7 | \( 1 + 7.74iT - 49T^{2} \) |
| 11 | \( 1 - 12.6iT - 121T^{2} \) |
| 13 | \( 1 + 15.4T + 169T^{2} \) |
| 17 | \( 1 - 25.2T + 289T^{2} \) |
| 19 | \( 1 - 8iT - 361T^{2} \) |
| 23 | \( 1 - 39.1iT - 529T^{2} \) |
| 29 | \( 1 + 24.4T + 841T^{2} \) |
| 31 | \( 1 + 7.74iT - 961T^{2} \) |
| 37 | \( 1 + 46.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 25.2T + 1.68e3T^{2} \) |
| 43 | \( 1 - 40iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 39.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 14.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + 25.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 15.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 80iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 10T + 5.32e3T^{2} \) |
| 79 | \( 1 - 54.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 139. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 50.5T + 7.92e3T^{2} \) |
| 97 | \( 1 - 50T + 9.40e3T^{2} \) |
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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
−9.424759648178483795375999913844, −8.004632510434120753887146032674, −7.39729892271878028220259292609, −6.92421844141348015758752495803, −5.70034330933159929750103833128, −5.21020115240814023913188878552, −4.18501907981897112050885192814, −3.28854647334688065658914211987, −2.03609073331094160235447200360, −1.27568988727627622868010805340,
0.38399446845455750695037565895, 1.87977628586547739495258776354, 2.63590930728273591662771393213, 3.51028939852078783971126718601, 4.98698665122142671185929163804, 5.49042226158809394782322093356, 6.11425978565099872504301118080, 6.99631886662816027563474481277, 7.991857242205535011612745588886, 8.795193599426796547981500347274
