| L(s) = 1 | + (−0.485 + 2.75i)5-s + (1.68 + 2.91i)7-s + (−0.258 + 0.447i)11-s + (−4.37 − 1.59i)13-s + (0.735 + 0.617i)17-s + (−3.12 + 3.04i)19-s + (−0.629 − 3.57i)23-s + (−2.63 − 0.958i)25-s + (−6.21 + 5.21i)29-s + (−2.38 − 4.13i)31-s + (−8.83 + 3.21i)35-s + 9.13·37-s + (6.54 − 2.38i)41-s + (−0.817 + 4.63i)43-s + (−10.4 + 8.74i)47-s + ⋯ |
| L(s) = 1 | + (−0.216 + 1.23i)5-s + (0.636 + 1.10i)7-s + (−0.0778 + 0.134i)11-s + (−1.21 − 0.441i)13-s + (0.178 + 0.149i)17-s + (−0.715 + 0.698i)19-s + (−0.131 − 0.744i)23-s + (−0.526 − 0.191i)25-s + (−1.15 + 0.968i)29-s + (−0.429 − 0.743i)31-s + (−1.49 + 0.543i)35-s + 1.50·37-s + (1.02 − 0.372i)41-s + (−0.124 + 0.706i)43-s + (−1.51 + 1.27i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.575 - 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.530484 + 1.02249i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.530484 + 1.02249i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (3.12 - 3.04i)T \) |
| good | 5 | \( 1 + (0.485 - 2.75i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-1.68 - 2.91i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.258 - 0.447i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (4.37 + 1.59i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.735 - 0.617i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (0.629 + 3.57i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (6.21 - 5.21i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (2.38 + 4.13i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 9.13T + 37T^{2} \) |
| 41 | \( 1 + (-6.54 + 2.38i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (0.817 - 4.63i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (10.4 - 8.74i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-1.20 - 6.81i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-10.9 - 9.19i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.05 - 6.00i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-3.38 + 2.84i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.66 + 9.42i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-5.65 + 2.05i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (9.85 - 3.58i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (2.39 + 4.14i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-8.30 - 3.02i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-8.46 - 7.10i)T + (16.8 + 95.5i)T^{2} \) |
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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
−10.85423641613247017534713117393, −10.00498439199056605839717631184, −9.103375513901357614364252762752, −7.992418922331131979312841369803, −7.42359831665845473045992885509, −6.29590065389037938142067511809, −5.48017719731709420764591624585, −4.31892153844660135924713506355, −2.93501310056892336600863798177, −2.15015519457790365016376344980,
0.59366379455735406325635346373, 2.03760663253561296552494120745, 3.85819895284241338656695765235, 4.66150209408773096195092201645, 5.34756455719911461518697971375, 6.83190163989672656962340099438, 7.65757062684222607501100996447, 8.362898220685349861918309222220, 9.373023991550634610995992379310, 10.03251183531345553209498852576
