| L(s) = 1 | + (−2.25 + 1.15i)2-s + (1.98 + 0.313i)3-s + (2.60 − 3.58i)4-s + (−0.867 + 2.06i)5-s + (−4.83 + 1.57i)6-s + (0.282 + 1.78i)7-s + (−0.962 + 6.07i)8-s + (0.969 + 0.315i)9-s + (−0.412 − 5.65i)10-s + (1.72 − 2.83i)11-s + (6.27 − 6.27i)12-s + (−1.02 − 2.00i)13-s + (−2.69 − 3.70i)14-s + (−2.36 + 3.80i)15-s + (−2.08 − 6.41i)16-s + (1.52 − 2.99i)17-s + ⋯ |
| L(s) = 1 | + (−1.59 + 0.813i)2-s + (1.14 + 0.181i)3-s + (1.30 − 1.79i)4-s + (−0.387 + 0.921i)5-s + (−1.97 + 0.641i)6-s + (0.106 + 0.674i)7-s + (−0.340 + 2.14i)8-s + (0.323 + 0.105i)9-s + (−0.130 − 1.78i)10-s + (0.519 − 0.854i)11-s + (1.81 − 1.81i)12-s + (−0.284 − 0.557i)13-s + (−0.720 − 0.991i)14-s + (−0.610 + 0.983i)15-s + (−0.521 − 1.60i)16-s + (0.370 − 0.726i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.260 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.260 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.451980 + 0.346231i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.451980 + 0.346231i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.867 - 2.06i)T \) |
| 11 | \( 1 + (-1.72 + 2.83i)T \) |
| good | 2 | \( 1 + (2.25 - 1.15i)T + (1.17 - 1.61i)T^{2} \) |
| 3 | \( 1 + (-1.98 - 0.313i)T + (2.85 + 0.927i)T^{2} \) |
| 7 | \( 1 + (-0.282 - 1.78i)T + (-6.65 + 2.16i)T^{2} \) |
| 13 | \( 1 + (1.02 + 2.00i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-1.52 + 2.99i)T + (-9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (-1.05 + 0.767i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (3.16 + 3.16i)T + 23iT^{2} \) |
| 29 | \( 1 + (-2.37 - 1.72i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (1.23 - 3.81i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-3.16 + 0.501i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (-0.766 - 1.05i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (4.55 - 4.55i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.20 - 7.62i)T + (-44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (-5.67 + 2.89i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (7.28 - 10.0i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (8.85 - 2.87i)T + (49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (2.46 - 2.46i)T - 67iT^{2} \) |
| 71 | \( 1 + (2.09 + 6.43i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-9.32 + 1.47i)T + (69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (0.353 - 1.08i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (9.32 + 4.75i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + 3.85iT - 89T^{2} \) |
| 97 | \( 1 + (-0.353 - 0.693i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−15.58912149340451778043905928331, −14.78303936392037444179931046578, −14.07909356382242773931211738125, −11.70122608063158167046526599025, −10.43763946655348412579333668762, −9.296463649793119154119773119816, −8.414578402384818030514113108938, −7.51535926649748935972309457339, −6.11277661535508098396782120705, −2.87748094991611074353617802411,
1.79196949988515795356777399372, 3.85431750368029900581463899505, 7.39485524022563727512692749591, 8.164561208035635685132543087183, 9.177339303926713878364629249910, 9.991973561768381093756175570361, 11.55397106039560084053484660544, 12.50112780081501836417546199188, 13.85097530229447740690031298933, 15.29638925491638915955134545682
