L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (2.54 − 0.717i)7-s + 0.999·8-s + (−5.09 − 2.94i)11-s − 4.05·13-s + (−1.89 − 1.84i)14-s + (−0.5 − 0.866i)16-s + (0.371 + 0.214i)17-s + (−5.30 + 3.06i)19-s + 5.88i·22-s + (−0.876 − 1.51i)23-s + (2.02 + 3.51i)26-s + (−0.651 + 2.56i)28-s + 0.0419i·29-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.962 − 0.271i)7-s + 0.353·8-s + (−1.53 − 0.886i)11-s − 1.12·13-s + (−0.506 − 0.493i)14-s + (−0.125 − 0.216i)16-s + (0.0901 + 0.0520i)17-s + (−1.21 + 0.703i)19-s + 1.25i·22-s + (−0.182 − 0.316i)23-s + (0.397 + 0.689i)26-s + (−0.123 + 0.484i)28-s + 0.00778i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.715 - 0.698i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.715 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8279025759\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8279025759\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.54 + 0.717i)T \) |
good | 11 | \( 1 + (5.09 + 2.94i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 4.05T + 13T^{2} \) |
| 17 | \( 1 + (-0.371 - 0.214i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (5.30 - 3.06i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.876 + 1.51i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 0.0419iT - 29T^{2} \) |
| 31 | \( 1 + (-7.92 - 4.57i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.928 + 0.536i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 8.61T + 41T^{2} \) |
| 43 | \( 1 - 11.0iT - 43T^{2} \) |
| 47 | \( 1 + (0.834 - 0.481i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.57 - 11.3i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.77 + 11.7i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.05 + 0.609i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-10.9 - 6.32i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2.54iT - 71T^{2} \) |
| 73 | \( 1 + (4.66 - 8.08i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.35 - 9.28i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 10.1iT - 83T^{2} \) |
| 89 | \( 1 + (3.15 + 5.46i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 2.59T + 97T^{2} \) |
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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
−8.553403154971262000408171068706, −8.104595643104299269007229362124, −7.68139197453259974306485983332, −6.58520103967957766471092421815, −5.56662470707408210884707960820, −4.80780645714587195592627813181, −4.13080810352585229543704563679, −2.84929315124293915180127352229, −2.30381143032202688272794875580, −1.00517070219244570131359075500,
0.32521830712250687671305067379, 2.05974087516458001665348294545, 2.55590249909829507351157577751, 4.27504565209181048809646460249, 4.88723181406656242715403005058, 5.41645057336213603542098899488, 6.42399365886341098691058878935, 7.35096464829947769213377544696, 7.77402866057765208488719557593, 8.406385102987290430929798440330