| L(s) = 1 | + (−1.18 − 1.56i)2-s + (1.30 + 1.43i)3-s + (−0.508 + 1.78i)4-s + (−1.37 + 3.55i)5-s + (0.700 − 3.74i)6-s + (−0.287 − 0.0266i)7-s + (−0.261 + 0.101i)8-s + (−0.0711 + 0.767i)9-s + (7.19 − 2.04i)10-s + (−4.70 − 1.33i)11-s + (−3.22 + 1.60i)12-s + (−4.36 + 1.68i)13-s + (0.298 + 0.482i)14-s + (−6.89 + 2.67i)15-s + (3.62 + 2.24i)16-s + (−0.136 + 4.12i)17-s + ⋯ |
| L(s) = 1 | + (−0.836 − 1.10i)2-s + (0.755 + 0.828i)3-s + (−0.254 + 0.893i)4-s + (−0.615 + 1.58i)5-s + (0.286 − 1.53i)6-s + (−0.108 − 0.0100i)7-s + (−0.0923 + 0.0357i)8-s + (−0.0237 + 0.255i)9-s + (2.27 − 0.647i)10-s + (−1.41 − 0.403i)11-s + (−0.932 + 0.464i)12-s + (−1.20 + 0.468i)13-s + (0.0798 + 0.128i)14-s + (−1.78 + 0.689i)15-s + (0.906 + 0.561i)16-s + (−0.0330 + 0.999i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0913 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0913 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.376364 + 0.412459i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.376364 + 0.412459i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (0.136 - 4.12i)T \) |
| good | 2 | \( 1 + (1.18 + 1.56i)T + (-0.547 + 1.92i)T^{2} \) |
| 3 | \( 1 + (-1.30 - 1.43i)T + (-0.276 + 2.98i)T^{2} \) |
| 5 | \( 1 + (1.37 - 3.55i)T + (-3.69 - 3.36i)T^{2} \) |
| 7 | \( 1 + (0.287 + 0.0266i)T + (6.88 + 1.28i)T^{2} \) |
| 11 | \( 1 + (4.70 + 1.33i)T + (9.35 + 5.79i)T^{2} \) |
| 13 | \( 1 + (4.36 - 1.68i)T + (9.60 - 8.75i)T^{2} \) |
| 19 | \( 1 + (-1.52 + 2.02i)T + (-5.19 - 18.2i)T^{2} \) |
| 23 | \( 1 + (5.38 + 0.499i)T + (22.6 + 4.22i)T^{2} \) |
| 29 | \( 1 + (-7.58 - 2.15i)T + (24.6 + 15.2i)T^{2} \) |
| 31 | \( 1 + (-1.86 - 4.81i)T + (-22.9 + 20.8i)T^{2} \) |
| 37 | \( 1 + (-7.62 - 3.79i)T + (22.2 + 29.5i)T^{2} \) |
| 41 | \( 1 + (-6.32 - 6.93i)T + (-3.78 + 40.8i)T^{2} \) |
| 43 | \( 1 + (-3.98 + 2.46i)T + (19.1 - 38.4i)T^{2} \) |
| 47 | \( 1 + (-0.466 - 5.03i)T + (-46.1 + 8.63i)T^{2} \) |
| 53 | \( 1 + (0.747 - 8.06i)T + (-52.0 - 9.73i)T^{2} \) |
| 59 | \( 1 + (-1.11 + 0.207i)T + (55.0 - 21.3i)T^{2} \) |
| 61 | \( 1 + (-1.06 + 5.68i)T + (-56.8 - 22.0i)T^{2} \) |
| 67 | \( 1 + (5.73 - 7.59i)T + (-18.3 - 64.4i)T^{2} \) |
| 71 | \( 1 + (11.4 + 1.05i)T + (69.7 + 13.0i)T^{2} \) |
| 73 | \( 1 + (-0.124 + 0.201i)T + (-32.5 - 65.3i)T^{2} \) |
| 79 | \( 1 + (-6.96 - 5.25i)T + (21.6 + 75.9i)T^{2} \) |
| 83 | \( 1 + (2.95 + 2.69i)T + (7.65 + 82.6i)T^{2} \) |
| 89 | \( 1 + (11.4 + 4.44i)T + (65.7 + 59.9i)T^{2} \) |
| 97 | \( 1 + (5.51 + 0.511i)T + (95.3 + 17.8i)T^{2} \) |
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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
−11.66442389969955066706922779478, −10.74821285819232750772195440163, −10.24034792567912115247000209205, −9.649517914869331551613123891821, −8.404840585672333076709052757527, −7.67684582215619492257007540117, −6.28447763836173952316176451828, −4.33803414882730730126287291193, −2.96574244306916734022973320254, −2.70356190175807334500147776165,
0.47326941233379922215373519273, 2.56531039115084704359940014567, 4.68129622588906809701870975034, 5.65657793345635092168273508755, 7.31235393682568686142452268143, 7.85656800785715905094842875617, 8.198023645569722157994755533476, 9.300408363326856082987018387928, 10.04655106991171316048924825422, 11.98077807676426081131054212802
