| L(s) = 1 | + (1.93 + 1.11i)5-s + (0.854 + 1.47i)7-s + (3.21 − 1.85i)11-s + (6.85 − 11.8i)13-s − 15i·17-s − 14.4·19-s + (−6.56 − 3.79i)23-s + (2.5 + 4.33i)25-s + (8.40 − 4.85i)29-s + (10.2 − 17.6i)31-s + 3.81i·35-s − 56.2·37-s + (−36.8 − 21.2i)41-s + (8.97 + 15.5i)43-s + (1.22 − 0.708i)47-s + ⋯ |
| L(s) = 1 | + (0.387 + 0.223i)5-s + (0.122 + 0.211i)7-s + (0.291 − 0.168i)11-s + (0.527 − 0.913i)13-s − 0.882i·17-s − 0.758·19-s + (−0.285 − 0.164i)23-s + (0.100 + 0.173i)25-s + (0.289 − 0.167i)29-s + (0.329 − 0.570i)31-s + 0.109i·35-s − 1.52·37-s + (−0.898 − 0.518i)41-s + (0.208 + 0.361i)43-s + (0.0260 − 0.0150i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.759274152\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.759274152\) |
| \(L(2)\) |
|
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 | \( 1 + (-1.93 - 1.11i)T \) |
| good | 7 | \( 1 + (-0.854 - 1.47i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-3.21 + 1.85i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-6.85 + 11.8i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 15iT - 289T^{2} \) |
| 19 | \( 1 + 14.4T + 361T^{2} \) |
| 23 | \( 1 + (6.56 + 3.79i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-8.40 + 4.85i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-10.2 + 17.6i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 56.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + (36.8 + 21.2i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-8.97 - 15.5i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-1.22 + 0.708i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 70.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-26.7 - 15.4i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-40.7 - 70.5i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (7 - 12.1i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 1.95iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 27.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + (6.74 + 11.6i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-64.3 + 37.1i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 123. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-5.70 - 9.88i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−8.852940648968295963603621519741, −8.436157535038519253894294372265, −7.38312745958260743882067464557, −6.58113429122790638860153550639, −5.77031110453312162503209588766, −5.01921819109504392010292548687, −3.87821943627917026263493401349, −2.91912333784811674514247236884, −1.88527447057827602977863593004, −0.48188617691692294171128384610,
1.27921123532902005630494551857, 2.13353095978684806082180039537, 3.55863562477775854908640898053, 4.33584925031645893936863333829, 5.25973250117059299434187643207, 6.31868624129681224553576417951, 6.76690451278177105254479799216, 7.907021994414003086462861016763, 8.699435049541528167560800429582, 9.233156303839297085654119876759
