| L(s) = 1 | + (−0.237 + 1.50i)2-s + (0.710 + 0.361i)3-s + (−0.295 − 0.0959i)4-s + (−1.71 − 1.43i)5-s + (−0.712 + 0.980i)6-s + (0.0869 + 0.170i)7-s + (−1.16 + 2.28i)8-s + (−1.38 − 1.91i)9-s + (2.56 − 2.23i)10-s + (1.77 − 2.80i)11-s + (−0.175 − 0.175i)12-s + (3.05 + 0.484i)13-s + (−0.276 + 0.0899i)14-s + (−0.698 − 1.63i)15-s + (−3.66 − 2.65i)16-s + (−3.66 + 0.579i)17-s + ⋯ |
| L(s) = 1 | + (−0.168 + 1.06i)2-s + (0.410 + 0.208i)3-s + (−0.147 − 0.0479i)4-s + (−0.766 − 0.641i)5-s + (−0.290 + 0.400i)6-s + (0.0328 + 0.0644i)7-s + (−0.412 + 0.808i)8-s + (−0.463 − 0.637i)9-s + (0.810 − 0.706i)10-s + (0.535 − 0.844i)11-s + (−0.0505 − 0.0505i)12-s + (0.847 + 0.134i)13-s + (−0.0739 + 0.0240i)14-s + (−0.180 − 0.423i)15-s + (−0.915 − 0.664i)16-s + (−0.887 + 0.140i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.319 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.319 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.686378 + 0.493054i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.686378 + 0.493054i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.71 + 1.43i)T \) |
| 11 | \( 1 + (-1.77 + 2.80i)T \) |
| good | 2 | \( 1 + (0.237 - 1.50i)T + (-1.90 - 0.618i)T^{2} \) |
| 3 | \( 1 + (-0.710 - 0.361i)T + (1.76 + 2.42i)T^{2} \) |
| 7 | \( 1 + (-0.0869 - 0.170i)T + (-4.11 + 5.66i)T^{2} \) |
| 13 | \( 1 + (-3.05 - 0.484i)T + (12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (3.66 - 0.579i)T + (16.1 - 5.25i)T^{2} \) |
| 19 | \( 1 + (-0.229 - 0.707i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (1.14 - 1.14i)T - 23iT^{2} \) |
| 29 | \( 1 + (2.95 - 9.07i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (0.283 - 0.206i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (4.81 - 2.45i)T + (21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (-6.36 + 2.06i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (3.72 + 3.72i)T + 43iT^{2} \) |
| 47 | \( 1 + (-5.61 + 11.0i)T + (-27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (1.41 - 8.91i)T + (-50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (-9.15 - 2.97i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (3.46 - 4.76i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-4.13 - 4.13i)T + 67iT^{2} \) |
| 71 | \( 1 + (-9.27 - 6.73i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-2.14 + 1.09i)T + (42.9 - 59.0i)T^{2} \) |
| 79 | \( 1 + (-0.542 + 0.394i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (2.60 + 16.4i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + 7.92iT - 89T^{2} \) |
| 97 | \( 1 + (1.36 + 0.215i)T + (92.2 + 29.9i)T^{2} \) |
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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
−15.64223678005457971085992101195, −14.79082013430984784238804762543, −13.68488695239633562581376148841, −12.06669704392461125776519582535, −11.13552817553577831567012186633, −8.768723831794898025084569732681, −8.661348894057334204091716597121, −7.01841093085232033338575205167, −5.66924853502106264560026071016, −3.68367733179799525995542556162,
2.40033684023239449929250978450, 3.98321059091092992012627531975, 6.58075031946221013134107698875, 7.945566117253026051992231013922, 9.412251293715021347171985683569, 10.83333663261056399294149800401, 11.41684266865311522881569135012, 12.61994507292337017696407819410, 13.85641317938197823534973907048, 15.09797627552240304524223808367
