L(s) = 1 | + (−0.5 + 0.866i)2-s + (1.18 + 1.26i)3-s + (−0.499 − 0.866i)4-s + (0.686 − 2.12i)5-s + (−1.68 + 0.396i)6-s + (2.5 + 0.866i)7-s + 0.999·8-s + (−0.186 + 2.99i)9-s + (1.5 + 1.65i)10-s + (0.813 − 0.469i)11-s + (0.500 − 1.65i)12-s + 2·13-s + (−2 + 1.73i)14-s + (3.5 − 1.65i)15-s + (−0.5 + 0.866i)16-s + (−5.74 + 3.31i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.684 + 0.728i)3-s + (−0.249 − 0.433i)4-s + (0.306 − 0.951i)5-s + (−0.688 + 0.161i)6-s + (0.944 + 0.327i)7-s + 0.353·8-s + (−0.0620 + 0.998i)9-s + (0.474 + 0.524i)10-s + (0.245 − 0.141i)11-s + (0.144 − 0.478i)12-s + 0.554·13-s + (−0.534 + 0.462i)14-s + (0.903 − 0.428i)15-s + (−0.125 + 0.216i)16-s + (−1.39 + 0.804i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 - 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.515 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.17663 + 0.665478i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.17663 + 0.665478i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (-1.18 - 1.26i)T \) |
| 5 | \( 1 + (-0.686 + 2.12i)T \) |
| 7 | \( 1 + (-2.5 - 0.866i)T \) |
good | 11 | \( 1 + (-0.813 + 0.469i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (5.74 - 3.31i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3 + 1.73i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.686 + 1.18i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 3.31iT - 29T^{2} \) |
| 31 | \( 1 + (-6.55 + 3.78i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (7.11 + 4.10i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 7.37T + 41T^{2} \) |
| 43 | \( 1 + 1.08iT - 43T^{2} \) |
| 47 | \( 1 + (7.37 + 4.25i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.18 - 3.78i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6.55 + 11.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (11.0 + 6.38i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.05 + 1.18i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 8.51iT - 71T^{2} \) |
| 73 | \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.55 - 7.89i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 11.8iT - 83T^{2} \) |
| 89 | \( 1 + (0.686 - 1.18i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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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
−12.78665339993164875945038740876, −11.27914470851601814860616866362, −10.44443307054541295888751043600, −9.101298713781177224477914644211, −8.714165136287436719251607465676, −7.946114334651656511953912994291, −6.27703381296364396863576202218, −5.01005213489748806786828141599, −4.20781049770100152978706835380, −1.95513667344711490004689839021,
1.69202001641479529898607694123, 2.88566775485323823750553858300, 4.32708014018596054336369281595, 6.34861727149625616978157756234, 7.28479412697082408594346692964, 8.281692311837272084729167176943, 9.187680073653195972287960208497, 10.37919991182282742146317803349, 11.24816106678372933623686090720, 12.02920218061278566476916354980