| L(s) = 1 | + (1.01 − 0.985i)2-s + (−0.0658 + 0.224i)3-s + (0.0574 − 1.99i)4-s + (−0.0287 − 2.23i)5-s + (0.154 + 0.292i)6-s + (−3.12 − 0.448i)7-s + (−1.91 − 2.08i)8-s + (2.47 + 1.59i)9-s + (−2.23 − 2.23i)10-s + (0.180 − 0.0824i)11-s + (0.444 + 0.144i)12-s + (−0.394 − 2.74i)13-s + (−3.60 + 2.62i)14-s + (0.502 + 0.140i)15-s + (−3.99 − 0.229i)16-s + (2.78 − 3.21i)17-s + ⋯ |
| L(s) = 1 | + (0.717 − 0.696i)2-s + (−0.0379 + 0.129i)3-s + (0.0287 − 0.999i)4-s + (−0.0128 − 0.999i)5-s + (0.0629 + 0.119i)6-s + (−1.17 − 0.169i)7-s + (−0.676 − 0.736i)8-s + (0.825 + 0.530i)9-s + (−0.706 − 0.708i)10-s + (0.0544 − 0.0248i)11-s + (0.128 + 0.0416i)12-s + (−0.109 − 0.760i)13-s + (−0.963 + 0.700i)14-s + (0.129 + 0.0363i)15-s + (−0.998 − 0.0573i)16-s + (0.675 − 0.780i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.166i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.134845 - 1.60407i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.134845 - 1.60407i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.01 + 0.985i)T \) |
| 5 | \( 1 + (0.0287 + 2.23i)T \) |
| 23 | \( 1 + (3.63 - 3.12i)T \) |
| good | 3 | \( 1 + (0.0658 - 0.224i)T + (-2.52 - 1.62i)T^{2} \) |
| 7 | \( 1 + (3.12 + 0.448i)T + (6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (-0.180 + 0.0824i)T + (7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (0.394 + 2.74i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (-2.78 + 3.21i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (0.0789 - 0.0683i)T + (2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (7.24 + 6.27i)T + (4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (0.0110 + 0.0375i)T + (-26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (0.475 - 0.739i)T + (-15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (1.70 - 1.09i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (3.81 + 1.12i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 - 4.76T + 47T^{2} \) |
| 53 | \( 1 + (-12.7 - 1.82i)T + (50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (0.150 + 1.04i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-9.52 + 2.79i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (-3.96 + 8.68i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-11.0 - 5.06i)T + (46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (1.33 - 1.15i)T + (10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (1.72 + 11.9i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-2.52 - 1.62i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-1.45 + 4.94i)T + (-74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (-11.1 + 7.13i)T + (40.2 - 88.2i)T^{2} \) |
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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
−9.809973646355281172511032822102, −9.319258518752353613413153469153, −7.978516564917209238240121256913, −7.03616461620179297556740530621, −5.82385494750072873674193149034, −5.23930024145421447533485289779, −4.16402072668383954442611758977, −3.41505957729930956056678188360, −2.03556308977575878432257300477, −0.58338452981220751514025298551,
2.25535533131212629280345210811, 3.55543472768309564252371624340, 3.94746877119948810906378002443, 5.50242580641601516189011652414, 6.39412126413899849805607605570, 6.84202442909375379928129758301, 7.54222516533885783933600701742, 8.728013713032356228831540032406, 9.664742709169089625695059522150, 10.36808401532468090929374123910
