| L(s) = 1 | − 2.23·5-s + 13.4i·7-s − 21.2i·11-s − 2.66·13-s + 19.7·17-s + 13.5i·19-s + 40.1i·23-s + 5.00·25-s + 1.41·29-s − 41.1i·31-s − 29.9i·35-s + 54.0·37-s + 2.74·41-s − 22.6i·43-s − 45.6i·47-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.91i·7-s − 1.92i·11-s − 0.204·13-s + 1.16·17-s + 0.713i·19-s + 1.74i·23-s + 0.200·25-s + 0.0488·29-s − 1.32i·31-s − 0.856i·35-s + 1.46·37-s + 0.0669·41-s − 0.527i·43-s − 0.971i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.626657054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.626657054\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23T \) |
| good | 7 | \( 1 - 13.4iT - 49T^{2} \) |
| 11 | \( 1 + 21.2iT - 121T^{2} \) |
| 13 | \( 1 + 2.66T + 169T^{2} \) |
| 17 | \( 1 - 19.7T + 289T^{2} \) |
| 19 | \( 1 - 13.5iT - 361T^{2} \) |
| 23 | \( 1 - 40.1iT - 529T^{2} \) |
| 29 | \( 1 - 1.41T + 841T^{2} \) |
| 31 | \( 1 + 41.1iT - 961T^{2} \) |
| 37 | \( 1 - 54.0T + 1.36e3T^{2} \) |
| 41 | \( 1 - 2.74T + 1.68e3T^{2} \) |
| 43 | \( 1 + 22.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 45.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 4.37T + 2.80e3T^{2} \) |
| 59 | \( 1 - 30.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 27.0T + 3.72e3T^{2} \) |
| 67 | \( 1 - 103. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 88.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 36.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 23.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 119. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 71.7T + 7.92e3T^{2} \) |
| 97 | \( 1 - 121.T + 9.40e3T^{2} \) |
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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
−8.710697846090961924955590710340, −8.123374965993028197288715592151, −7.53298764433036245715626864045, −6.13217008260357791902296800864, −5.75038460209289919497744705506, −5.25863199466849165302502933109, −3.79519631532260069976255508318, −3.17296833900208914609627514929, −2.29416091351185529859069078707, −0.963825416609322125332781521643,
0.46010018962126162496495046168, 1.42036618773608169281254250436, 2.72560835487349137527605073492, 3.77828786216903460571410926003, 4.59253882514582027195084725183, 4.84723646938694233376475453833, 6.48982692462351737407678513121, 6.95883560264258503537649967699, 7.64410572848923957167238196238, 8.074152110770481582808255107982
