| L(s) = 1 | + 12.3i·5-s − 21.4i·7-s + 15.5·11-s − 2·13-s + 24.7i·17-s − 85.6i·19-s + 124.·23-s − 27.9·25-s − 272. i·29-s + 278. i·31-s + 265.·35-s + 128·37-s + 296. i·41-s + 42.8i·43-s + 592.·47-s + ⋯ |
| L(s) = 1 | + 1.10i·5-s − 1.15i·7-s + 0.427·11-s − 0.0426·13-s + 0.352i·17-s − 1.03i·19-s + 1.13·23-s − 0.223·25-s − 1.74i·29-s + 1.61i·31-s + 1.27·35-s + 0.568·37-s + 1.13i·41-s + 0.151i·43-s + 1.83·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.966235641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.966235641\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 12.3iT - 125T^{2} \) |
| 7 | \( 1 + 21.4iT - 343T^{2} \) |
| 11 | \( 1 - 15.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 24.7iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 85.6iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 124.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 272. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 278. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 128T + 5.06e4T^{2} \) |
| 41 | \( 1 - 296. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 42.8iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 592.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 259. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 530.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 340T + 2.26e5T^{2} \) |
| 67 | \( 1 + 899. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 966.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 817T + 3.89e5T^{2} \) |
| 79 | \( 1 + 214. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 358.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 915. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 965T + 9.12e5T^{2} \) |
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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
−10.82370914627880835951665216622, −10.01667778252886652540537215578, −9.012749844495660707239766134894, −7.74348604286362851974922311795, −6.96890865900751892669119750201, −6.32776547674155929867097612868, −4.77970935466323346122811053489, −3.71077377103348988559483492146, −2.63256536190679395914160410316, −0.882808184468734170764575946873,
0.985108184334498196229848611196, 2.36429633689710327201666054167, 3.86356639685480976218179485039, 5.12676881306033789701585258485, 5.72377422155413186758509654585, 7.02045391583874206410830124666, 8.246120336317813008287759095245, 8.969926459462073822821594599212, 9.510579436568969680058508012091, 10.81242812845550204191093339554
