L(s) = 1 | + (0.707 − 0.707i)2-s + (1.08 + 1.35i)3-s − 1.00i·4-s + (0.707 + 0.707i)5-s + (1.72 + 0.189i)6-s − 3.77·7-s + (−0.707 − 0.707i)8-s + (−0.651 + 2.92i)9-s + 1.00·10-s − 1.81·11-s + (1.35 − 1.08i)12-s + (−3.90 + 3.90i)13-s + (−2.67 + 2.67i)14-s + (−0.189 + 1.72i)15-s − 1.00·16-s + (1.04 + 1.04i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (0.625 + 0.780i)3-s − 0.500i·4-s + (0.316 + 0.316i)5-s + (0.702 + 0.0772i)6-s − 1.42·7-s + (−0.250 − 0.250i)8-s + (−0.217 + 0.976i)9-s + 0.316·10-s − 0.547·11-s + (0.390 − 0.312i)12-s + (−1.08 + 1.08i)13-s + (−0.713 + 0.713i)14-s + (−0.0488 + 0.444i)15-s − 0.250·16-s + (0.253 + 0.253i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.254 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.254 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.556590580\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.556590580\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + (-1.08 - 1.35i)T \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
| 37 | \( 1 + (-5.83 + 1.70i)T \) |
good | 7 | \( 1 + 3.77T + 7T^{2} \) |
| 11 | \( 1 + 1.81T + 11T^{2} \) |
| 13 | \( 1 + (3.90 - 3.90i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.04 - 1.04i)T + 17iT^{2} \) |
| 19 | \( 1 + (1.01 - 1.01i)T - 19iT^{2} \) |
| 23 | \( 1 + (-5.79 - 5.79i)T + 23iT^{2} \) |
| 29 | \( 1 + (3.80 - 3.80i)T - 29iT^{2} \) |
| 31 | \( 1 + (-4.46 - 4.46i)T + 31iT^{2} \) |
| 41 | \( 1 + 9.74T + 41T^{2} \) |
| 43 | \( 1 + (-6.17 + 6.17i)T - 43iT^{2} \) |
| 47 | \( 1 + 7.59iT - 47T^{2} \) |
| 53 | \( 1 - 1.46iT - 53T^{2} \) |
| 59 | \( 1 + (-3.27 - 3.27i)T + 59iT^{2} \) |
| 61 | \( 1 + (0.251 + 0.251i)T + 61iT^{2} \) |
| 67 | \( 1 + 1.79iT - 67T^{2} \) |
| 71 | \( 1 + 14.5iT - 71T^{2} \) |
| 73 | \( 1 + 7.13iT - 73T^{2} \) |
| 79 | \( 1 + (8.05 - 8.05i)T - 79iT^{2} \) |
| 83 | \( 1 + 7.15iT - 83T^{2} \) |
| 89 | \( 1 + (7.47 - 7.47i)T - 89iT^{2} \) |
| 97 | \( 1 + (-3.22 + 3.22i)T - 97iT^{2} \) |
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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
−10.09140387720433953937506454523, −9.426390181454429611375091808003, −8.896255538645529613153036979020, −7.45727859434552358657239914962, −6.72595909144769867889491403988, −5.59699489096664758995242416680, −4.80013706665312180249726947811, −3.64930475069839510137225512373, −3.03535450224015494797060146357, −2.06555016519168386470581259567,
0.49409450069612180244613679894, 2.65640182293147885663048228931, 2.95323733701973077248336213848, 4.38631865537857769891681496087, 5.53337907063245440129808837897, 6.30592132329250796717982934520, 7.05826513894475522876078172367, 7.81735646234241199180639892442, 8.604193527658009431603710278016, 9.601262799068371443688160468078