| L(s) = 1 | + (−0.613 + 0.223i)3-s + (−0.233 + 1.32i)5-s + (0.766 + 1.32i)7-s + (−1.97 + 1.65i)9-s + (−0.592 + 1.02i)11-s + (−2.55 − 0.929i)13-s + (−0.152 − 0.866i)15-s + (2.97 + 2.49i)17-s + (−0.819 + 4.28i)19-s + (−0.766 − 0.642i)21-s + (0.879 + 4.98i)23-s + (2.99 + 1.08i)25-s + (1.81 − 3.15i)27-s + (−3.56 + 2.99i)29-s + (−1.91 − 3.32i)31-s + ⋯ |
| L(s) = 1 | + (−0.354 + 0.128i)3-s + (−0.104 + 0.593i)5-s + (0.289 + 0.501i)7-s + (−0.657 + 0.551i)9-s + (−0.178 + 0.309i)11-s + (−0.708 − 0.257i)13-s + (−0.0394 − 0.223i)15-s + (0.720 + 0.604i)17-s + (−0.187 + 0.982i)19-s + (−0.167 − 0.140i)21-s + (0.183 + 1.03i)23-s + (0.598 + 0.217i)25-s + (0.350 − 0.606i)27-s + (−0.661 + 0.555i)29-s + (−0.344 − 0.596i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.174 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.174 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.601762 + 0.717628i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.601762 + 0.717628i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (0.819 - 4.28i)T \) |
| good | 3 | \( 1 + (0.613 - 0.223i)T + (2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (0.233 - 1.32i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.766 - 1.32i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.592 - 1.02i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.55 + 0.929i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.97 - 2.49i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.879 - 4.98i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (3.56 - 2.99i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (1.91 + 3.32i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 4.10T + 37T^{2} \) |
| 41 | \( 1 + (-9.38 + 3.41i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.51 + 8.57i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (0.439 - 0.368i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.511 - 2.89i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-3.01 - 2.52i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (0.784 + 4.44i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-2.97 + 2.49i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.20 + 6.83i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (5.75 - 2.09i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-9.21 + 3.35i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-6.15 - 10.6i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.27 - 0.829i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-5.64 - 4.73i)T + (16.8 + 95.5i)T^{2} \) |
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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
−11.95517598558279916337527509552, −10.95286612362174477026788420779, −10.35215792468918431765633084927, −9.207895414677508441437969787808, −8.038548353622122823968993792741, −7.26499829692641123948169479587, −5.82106117924257123185685988122, −5.19299902389590848596557018842, −3.58340430887794600867611002422, −2.19256869397601100123996133621,
0.71901188043701531708829781776, 2.84084348296607215372602000614, 4.43402312012843698946608803679, 5.34890992854053397536350559769, 6.56283176082568778730590435577, 7.58065590967765258454181295276, 8.680021609815855287316862324803, 9.476483381287197201158985531086, 10.70677369404668015900949249863, 11.49051800693761865436063566181
