| L(s) = 1 | + (−0.891 − 0.453i)2-s + (0.587 + 0.809i)4-s + (−2.53 − 0.402i)5-s + (1.91 − 0.150i)7-s + (−0.156 − 0.987i)8-s + (2.07 + 1.51i)10-s + (−0.481 + 2.00i)11-s + (−1.04 + 0.889i)13-s + (−1.77 − 0.734i)14-s + (−0.309 + 0.951i)16-s + (−2.39 − 3.91i)17-s + (−2.79 − 2.38i)19-s + (−1.16 − 2.29i)20-s + (1.33 − 1.56i)22-s + (1.05 + 3.24i)23-s + ⋯ |
| L(s) = 1 | + (−0.630 − 0.321i)2-s + (0.293 + 0.404i)4-s + (−1.13 − 0.179i)5-s + (0.723 − 0.0569i)7-s + (−0.0553 − 0.349i)8-s + (0.657 + 0.477i)10-s + (−0.145 + 0.604i)11-s + (−0.288 + 0.246i)13-s + (−0.473 − 0.196i)14-s + (−0.0772 + 0.237i)16-s + (−0.581 − 0.948i)17-s + (−0.640 − 0.547i)19-s + (−0.260 − 0.512i)20-s + (0.285 − 0.334i)22-s + (0.219 + 0.676i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 - 0.570i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.821 - 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0339944 + 0.108587i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0339944 + 0.108587i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.891 + 0.453i)T \) |
| 3 | \( 1 \) |
| 41 | \( 1 + (6.13 + 1.82i)T \) |
| good | 5 | \( 1 + (2.53 + 0.402i)T + (4.75 + 1.54i)T^{2} \) |
| 7 | \( 1 + (-1.91 + 0.150i)T + (6.91 - 1.09i)T^{2} \) |
| 11 | \( 1 + (0.481 - 2.00i)T + (-9.80 - 4.99i)T^{2} \) |
| 13 | \( 1 + (1.04 - 0.889i)T + (2.03 - 12.8i)T^{2} \) |
| 17 | \( 1 + (2.39 + 3.91i)T + (-7.71 + 15.1i)T^{2} \) |
| 19 | \( 1 + (2.79 + 2.38i)T + (2.97 + 18.7i)T^{2} \) |
| 23 | \( 1 + (-1.05 - 3.24i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (2.25 - 3.68i)T + (-13.1 - 25.8i)T^{2} \) |
| 31 | \( 1 + (-1.46 + 2.01i)T + (-9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (7.99 - 5.80i)T + (11.4 - 35.1i)T^{2} \) |
| 43 | \( 1 + (4.39 - 8.62i)T + (-25.2 - 34.7i)T^{2} \) |
| 47 | \( 1 + (-0.0235 + 0.299i)T + (-46.4 - 7.35i)T^{2} \) |
| 53 | \( 1 + (-0.335 - 0.205i)T + (24.0 + 47.2i)T^{2} \) |
| 59 | \( 1 + (3.35 - 1.08i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-1.10 + 0.560i)T + (35.8 - 49.3i)T^{2} \) |
| 67 | \( 1 + (-0.716 - 2.98i)T + (-59.6 + 30.4i)T^{2} \) |
| 71 | \( 1 + (10.1 + 2.44i)T + (63.2 + 32.2i)T^{2} \) |
| 73 | \( 1 + (7.20 + 7.20i)T + 73iT^{2} \) |
| 79 | \( 1 + (-0.632 + 0.261i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 - 12.6iT - 83T^{2} \) |
| 89 | \( 1 + (-0.333 - 4.23i)T + (-87.9 + 13.9i)T^{2} \) |
| 97 | \( 1 + (3.33 - 0.800i)T + (86.4 - 44.0i)T^{2} \) |
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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
−10.83716117433887751725651807899, −9.838409677730163844884368868603, −8.935212872065621716385994668826, −8.194260148411914923558320192705, −7.41789991409591159650363683495, −6.75843634137775055979166723118, −5.02486520961817606459051826045, −4.35021221343232249606961984716, −3.09088681808607809869661447952, −1.70660895929001208016327177847,
0.07009057009152331127360264235, 1.92201736228014268217310091599, 3.48493206887271878385885151575, 4.51990928277837973772961695764, 5.66976734835009924675585339870, 6.72280651627015559635177144123, 7.62152548431384418865171245564, 8.366851409086904546246681099449, 8.757093142485617112958792341176, 10.22520174713018852854226776515
