| L(s) = 1 | + 6.03·5-s − 11.8i·7-s − 6.10i·11-s − 14.8·13-s + 26.6·17-s − 9.45i·19-s − 19.9i·23-s + 11.4·25-s − 44.6·29-s + 6.26i·31-s − 71.3i·35-s − 6.65·37-s − 17.6·41-s − 23.4i·43-s + 42.0i·47-s + ⋯ |
| L(s) = 1 | + 1.20·5-s − 1.68i·7-s − 0.554i·11-s − 1.14·13-s + 1.57·17-s − 0.497i·19-s − 0.866i·23-s + 0.456·25-s − 1.53·29-s + 0.202i·31-s − 2.03i·35-s − 0.179·37-s − 0.430·41-s − 0.544i·43-s + 0.894i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.915507881\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.915507881\) |
| \(L(2)\) |
|
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 | \( 1 - 6.03T + 25T^{2} \) |
| 7 | \( 1 + 11.8iT - 49T^{2} \) |
| 11 | \( 1 + 6.10iT - 121T^{2} \) |
| 13 | \( 1 + 14.8T + 169T^{2} \) |
| 17 | \( 1 - 26.6T + 289T^{2} \) |
| 19 | \( 1 + 9.45iT - 361T^{2} \) |
| 23 | \( 1 + 19.9iT - 529T^{2} \) |
| 29 | \( 1 + 44.6T + 841T^{2} \) |
| 31 | \( 1 - 6.26iT - 961T^{2} \) |
| 37 | \( 1 + 6.65T + 1.36e3T^{2} \) |
| 41 | \( 1 + 17.6T + 1.68e3T^{2} \) |
| 43 | \( 1 + 23.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 42.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 51.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + 37.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 90.7T + 3.72e3T^{2} \) |
| 67 | \( 1 - 61.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 39.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 35.0T + 5.32e3T^{2} \) |
| 79 | \( 1 - 90.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 118. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 14.4T + 7.92e3T^{2} \) |
| 97 | \( 1 + 135.T + 9.40e3T^{2} \) |
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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
−9.588711788159269348600655548848, −8.315309747527154937277969066041, −7.42109509382134042172135304041, −6.84513959787006106607853634979, −5.79149768831244843040170609979, −5.04807683373740212707211248025, −3.97398418453739249048627704158, −2.93965511554995307021103141314, −1.66120937248628843014309568886, −0.51733113287391034592423169345,
1.71066401585566755670032819574, 2.33702073149281850025354189657, 3.43088431818682554299662324653, 5.09749419420438512399611123784, 5.49433951303712749139040601718, 6.16359438780180400015395771795, 7.33429193188848630407757226941, 8.130595260426988200758023037284, 9.259479062807137572566762629222, 9.620894933952847876619675803126
