| L(s) = 1 | + 7.67·5-s + 7.21·7-s − 6.05·11-s + 2.29i·13-s − 21.8i·17-s + 34.8i·19-s − 21.5i·23-s + 33.8·25-s + 10.9·29-s + 37.6·31-s + 55.3·35-s + 34.8i·37-s + 13.3i·41-s − 60.5i·43-s + 3.34i·47-s + ⋯ |
| L(s) = 1 | + 1.53·5-s + 1.03·7-s − 0.550·11-s + 0.176i·13-s − 1.28i·17-s + 1.83i·19-s − 0.935i·23-s + 1.35·25-s + 0.376·29-s + 1.21·31-s + 1.58·35-s + 0.941i·37-s + 0.325i·41-s − 1.40i·43-s + 0.0712i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00404i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.00404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.24232 + 0.00453585i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.24232 + 0.00453585i\) |
| \(L(2)\) |
|
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 | \( 1 - 7.67T + 25T^{2} \) |
| 7 | \( 1 - 7.21T + 49T^{2} \) |
| 11 | \( 1 + 6.05T + 121T^{2} \) |
| 13 | \( 1 - 2.29iT - 169T^{2} \) |
| 17 | \( 1 + 21.8iT - 289T^{2} \) |
| 19 | \( 1 - 34.8iT - 361T^{2} \) |
| 23 | \( 1 + 21.5iT - 529T^{2} \) |
| 29 | \( 1 - 10.9T + 841T^{2} \) |
| 31 | \( 1 - 37.6T + 961T^{2} \) |
| 37 | \( 1 - 34.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 13.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 60.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 3.34iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 35.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 37.1T + 3.48e3T^{2} \) |
| 61 | \( 1 - 25.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 25.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 37.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 77.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + 31.3T + 6.24e3T^{2} \) |
| 83 | \( 1 + 55.3T + 6.88e3T^{2} \) |
| 89 | \( 1 + 5.43iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 52.8T + 9.40e3T^{2} \) |
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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
−11.60088514830199889628648892377, −10.39249287843133203400256985202, −9.920978892900802600508266816831, −8.747494593214243341644964935521, −7.82142618228697493399681939400, −6.49799547089217442382159187023, −5.52194871486418810310681208917, −4.64538288003719821493114741381, −2.68114372964456075806356629471, −1.50658049238568277020740302238,
1.50250572812226948328656156600, 2.68171641774824260029629214425, 4.63776127653140892912084347502, 5.52876129074426265857785487610, 6.49436256805519652325869479594, 7.80279611109657617109039565538, 8.805633763558822401020390116385, 9.743722869729564495365374446317, 10.64301083876033146533583984550, 11.37471173556774054076158442711
