L(s) = 1 | + (−2.12 + 1.22i)2-s + (1.35 − 0.780i)3-s + (2.01 − 3.49i)4-s + (−1.45 − 1.70i)5-s + (−1.91 + 3.32i)6-s − 4.50i·7-s + 4.99i·8-s + (−0.281 + 0.487i)9-s + (5.17 + 1.83i)10-s + 2.19·11-s − 6.29i·12-s + (3.25 + 1.87i)13-s + (5.53 + 9.58i)14-s + (−3.29 − 1.16i)15-s + (−2.09 − 3.63i)16-s + (−0.576 + 0.332i)17-s + ⋯ |
L(s) = 1 | + (−1.50 + 0.868i)2-s + (0.780 − 0.450i)3-s + (1.00 − 1.74i)4-s + (−0.649 − 0.760i)5-s + (−0.782 + 1.35i)6-s − 1.70i·7-s + 1.76i·8-s + (−0.0938 + 0.162i)9-s + (1.63 + 0.580i)10-s + 0.662·11-s − 1.81i·12-s + (0.902 + 0.521i)13-s + (1.47 + 2.56i)14-s + (−0.849 − 0.300i)15-s + (−0.524 − 0.909i)16-s + (−0.139 + 0.0807i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 + 0.516i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.559269 - 0.155579i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.559269 - 0.155579i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (1.45 + 1.70i)T \) |
| 19 | \( 1 + (3.79 + 2.13i)T \) |
good | 2 | \( 1 + (2.12 - 1.22i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.35 + 0.780i)T + (1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + 4.50iT - 7T^{2} \) |
| 11 | \( 1 - 2.19T + 11T^{2} \) |
| 13 | \( 1 + (-3.25 - 1.87i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.576 - 0.332i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-0.422 - 0.244i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.79 + 3.11i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 6.83T + 31T^{2} \) |
| 37 | \( 1 - 3.01iT - 37T^{2} \) |
| 41 | \( 1 + (0.0362 + 0.0627i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.364 + 0.210i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.34 - 2.51i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.26 + 1.30i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.26 - 10.8i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.53 - 6.11i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.95 - 2.86i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.48 + 6.03i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.56 - 1.47i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.66 - 9.81i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 15.6iT - 83T^{2} \) |
| 89 | \( 1 + (0.668 - 1.15i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.79 - 2.19i)T + (48.5 - 84.0i)T^{2} \) |
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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
−14.00982225422252755646900137426, −13.22273748165101071711365987757, −11.38347362485622197691465943731, −10.36555354937420577152610354418, −9.025195137228160796138962251129, −8.321715592165311815444631479575, −7.48052057758237673622973848498, −6.55153383213725490929494879866, −4.19964016437422200919886915340, −1.16896047195888895046040648865,
2.48362557078448155457709764839, 3.53558100298402372647174399946, 6.37958602823041773154440623440, 8.198109972309405239677349916334, 8.654769067658833810330665584810, 9.578462255975442074894943705506, 10.73356541426212926751357042533, 11.69945778389984164989817206861, 12.42358338087401937754641670510, 14.39920537041960389356487361450