| L(s) = 1 | − 5.60i·3-s + 2.23·5-s − 1.32i·7-s − 22.4·9-s − 11.2i·11-s − 17.4·13-s − 12.5i·15-s + 18·17-s − 5.29i·19-s − 7.41·21-s + 15.1i·23-s + 5.00·25-s + 75.1i·27-s − 8.83·29-s − 42.1i·31-s + ⋯ |
| L(s) = 1 | − 1.86i·3-s + 0.447·5-s − 0.189i·7-s − 2.49·9-s − 1.01i·11-s − 1.33·13-s − 0.835i·15-s + 1.05·17-s − 0.278i·19-s − 0.353·21-s + 0.659i·23-s + 0.200·25-s + 2.78i·27-s − 0.304·29-s − 1.36i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.25102i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25102i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23T \) |
| good | 3 | \( 1 + 5.60iT - 9T^{2} \) |
| 7 | \( 1 + 1.32iT - 49T^{2} \) |
| 11 | \( 1 + 11.2iT - 121T^{2} \) |
| 13 | \( 1 + 17.4T + 169T^{2} \) |
| 17 | \( 1 - 18T + 289T^{2} \) |
| 19 | \( 1 + 5.29iT - 361T^{2} \) |
| 23 | \( 1 - 15.1iT - 529T^{2} \) |
| 29 | \( 1 + 8.83T + 841T^{2} \) |
| 31 | \( 1 + 42.1iT - 961T^{2} \) |
| 37 | \( 1 + 33.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 28.2T + 1.68e3T^{2} \) |
| 43 | \( 1 - 25.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 10.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 28.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 44.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 77.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 36.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 97.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 15.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + 112. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 92.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 59.6T + 7.92e3T^{2} \) |
| 97 | \( 1 - 108.T + 9.40e3T^{2} \) |
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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
−11.31923778340501274339089432230, −9.993485680628178555822629147348, −8.831770886625032014024746607818, −7.79876040426896312231135348081, −7.20762674324186764795211622504, −6.12562135076906884646180380720, −5.34401460526845711408992377623, −3.14330157745783462108446927244, −1.96389670856562315505537350462, −0.57159260950025737000280026094,
2.50509635212001684715592164834, 3.78769037911237320577150383285, 4.94603581459413221620684553329, 5.45996319226312749054783557262, 7.03837170044908272976522040656, 8.453229500447373648226525639328, 9.382945366025510900990908997178, 10.11760486430763857008537271161, 10.44761989177309909731739779478, 11.83145227272525659125847593630
