L(s) = 1 | + (−0.968 − 0.659i)2-s + (0.630 + 0.584i)3-s + (−0.229 − 0.583i)4-s + (3.03 − 0.936i)5-s + (−0.224 − 0.982i)6-s + (−2.50 + 0.844i)7-s + (−0.684 + 3.00i)8-s + (−0.168 − 2.25i)9-s + (−3.55 − 1.09i)10-s + (−0.358 + 4.78i)11-s + (0.197 − 0.502i)12-s + (−3.44 + 1.65i)13-s + (2.98 + 0.836i)14-s + (2.46 + 1.18i)15-s + (1.72 − 1.59i)16-s + (0.602 − 0.0908i)17-s + ⋯ |
L(s) = 1 | + (−0.684 − 0.466i)2-s + (0.363 + 0.337i)3-s + (−0.114 − 0.291i)4-s + (1.35 − 0.418i)5-s + (−0.0915 − 0.400i)6-s + (−0.947 + 0.319i)7-s + (−0.242 + 1.06i)8-s + (−0.0563 − 0.751i)9-s + (−1.12 − 0.346i)10-s + (−0.108 + 1.44i)11-s + (0.0568 − 0.144i)12-s + (−0.955 + 0.460i)13-s + (0.797 + 0.223i)14-s + (0.635 + 0.305i)15-s + (0.430 − 0.399i)16-s + (0.146 − 0.0220i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 + 0.573i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.656378 - 0.206949i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.656378 - 0.206949i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (2.50 - 0.844i)T \) |
good | 2 | \( 1 + (0.968 + 0.659i)T + (0.730 + 1.86i)T^{2} \) |
| 3 | \( 1 + (-0.630 - 0.584i)T + (0.224 + 2.99i)T^{2} \) |
| 5 | \( 1 + (-3.03 + 0.936i)T + (4.13 - 2.81i)T^{2} \) |
| 11 | \( 1 + (0.358 - 4.78i)T + (-10.8 - 1.63i)T^{2} \) |
| 13 | \( 1 + (3.44 - 1.65i)T + (8.10 - 10.1i)T^{2} \) |
| 17 | \( 1 + (-0.602 + 0.0908i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (1.98 + 3.43i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.22 - 0.486i)T + (21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (0.769 + 0.965i)T + (-6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (-0.607 + 1.05i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.77 - 7.07i)T + (-27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (-1.52 + 6.66i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (-0.100 - 0.439i)T + (-38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (5.92 + 4.04i)T + (17.1 + 43.7i)T^{2} \) |
| 53 | \( 1 + (-0.873 - 2.22i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (-3.38 - 1.04i)T + (48.7 + 33.2i)T^{2} \) |
| 61 | \( 1 + (-4.53 + 11.5i)T + (-44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (-1.61 + 2.78i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.01 - 6.29i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (-2.14 + 1.45i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (2.76 + 4.79i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.63 - 1.27i)T + (51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (-0.623 - 8.32i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 - 6.49T + 97T^{2} \) |
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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
−15.36378042725775037772710110698, −14.51179834043876496499623580813, −13.19607487213481000906458085349, −12.08506890883546630106506981927, −10.14818212600568737206435617503, −9.586644823742727031322313030340, −9.005173385777941671303412729947, −6.62121983070430960727250225797, −5.05289993446727918767253369310, −2.31367542381274949372315777704,
2.98385116821430465205076926101, 5.87624500497358214110217519885, 7.13739021528771034543487620064, 8.431671513665082207879619365142, 9.654379970155721774314448618822, 10.58123773484093621659543453302, 12.78592081737007342241226953391, 13.44431536995183012668148973981, 14.45625867274441921910644581930, 16.25892309100275763905753452861