| L(s) = 1 | + (0.891 + 0.453i)2-s + (0.368 + 1.69i)3-s + (0.587 + 0.809i)4-s + (2.21 − 0.314i)5-s + (−0.439 + 1.67i)6-s + (−2.72 − 2.72i)7-s + (0.156 + 0.987i)8-s + (−2.72 + 1.24i)9-s + (2.11 + 0.725i)10-s + (−0.335 + 0.109i)11-s + (−1.15 + 1.29i)12-s + (−1.12 − 2.20i)13-s + (−1.19 − 3.66i)14-s + (1.34 + 3.63i)15-s + (−0.309 + 0.951i)16-s + (3.49 − 0.554i)17-s + ⋯ |
| L(s) = 1 | + (0.630 + 0.321i)2-s + (0.212 + 0.977i)3-s + (0.293 + 0.404i)4-s + (0.990 − 0.140i)5-s + (−0.179 + 0.683i)6-s + (−1.03 − 1.03i)7-s + (0.0553 + 0.349i)8-s + (−0.909 + 0.416i)9-s + (0.668 + 0.229i)10-s + (−0.101 + 0.0328i)11-s + (−0.332 + 0.373i)12-s + (−0.312 − 0.612i)13-s + (−0.318 − 0.980i)14-s + (0.348 + 0.937i)15-s + (−0.0772 + 0.237i)16-s + (0.848 − 0.134i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.44123 + 0.816046i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44123 + 0.816046i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.368 - 1.69i)T \) |
| 5 | \( 1 + (-2.21 + 0.314i)T \) |
| good | 7 | \( 1 + (2.72 + 2.72i)T + 7iT^{2} \) |
| 11 | \( 1 + (0.335 - 0.109i)T + (8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (1.12 + 2.20i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-3.49 + 0.554i)T + (16.1 - 5.25i)T^{2} \) |
| 19 | \( 1 + (3.84 - 5.29i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-3.55 + 6.98i)T + (-13.5 - 18.6i)T^{2} \) |
| 29 | \( 1 + (5.05 - 3.67i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (3.39 + 2.46i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-4.33 + 2.20i)T + (21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (-8.06 - 2.62i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (5.16 - 5.16i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.668 - 4.22i)T + (-44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (4.34 + 0.688i)T + (50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (-0.713 + 2.19i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-0.0451 - 0.139i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-1.18 - 7.47i)T + (-63.7 + 20.7i)T^{2} \) |
| 71 | \( 1 + (-3.62 - 4.98i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (9.30 + 4.74i)T + (42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (-0.803 - 1.10i)T + (-24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-0.915 - 5.77i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (-0.633 - 1.94i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-6.93 - 1.09i)T + (92.2 + 29.9i)T^{2} \) |
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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
−13.13498335182745402283962962123, −12.68524482401986158407652047618, −10.83860679668886260328506241715, −10.17010154342876895905917026063, −9.318068077048999083191731035807, −7.893518413388684547807548901058, −6.44865002375064244299419173593, −5.45530019074201416697771637798, −4.14974105053434421454692567805, −2.92100875086373541782612110389,
2.06059582230251602149373681836, 3.12938908539832947728762731045, 5.42233884322685825819149553218, 6.22332606807659111715946862825, 7.19462254627743664938376974161, 8.990196176952915016705800921145, 9.616769963186176455646440353018, 11.13163009471834566410892797479, 12.16495586362122767158753145945, 13.01723784842085085742612327054
