L(s) = 1 | − 1.73·3-s − 3i·5-s + 3.46i·7-s + 2.99·9-s − 1.73·11-s − 5·13-s + 5.19i·15-s + i·17-s − 1.73i·19-s − 5.99i·21-s + 5.19·23-s − 4·25-s − 5.19·27-s + 6i·29-s + 6.92i·31-s + ⋯ |
L(s) = 1 | − 1.00·3-s − 1.34i·5-s + 1.30i·7-s + 0.999·9-s − 0.522·11-s − 1.38·13-s + 1.34i·15-s + 0.242i·17-s − 0.397i·19-s − 1.30i·21-s + 1.08·23-s − 0.800·25-s − 1.00·27-s + 1.11i·29-s + 1.24i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.431572 + 0.431572i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.431572 + 0.431572i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73T \) |
| 17 | \( 1 - iT \) |
good | 5 | \( 1 + 3iT - 5T^{2} \) |
| 7 | \( 1 - 3.46iT - 7T^{2} \) |
| 11 | \( 1 + 1.73T + 11T^{2} \) |
| 13 | \( 1 + 5T + 13T^{2} \) |
| 19 | \( 1 + 1.73iT - 19T^{2} \) |
| 23 | \( 1 - 5.19T + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 6.92iT - 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 3iT - 41T^{2} \) |
| 43 | \( 1 - 12.1iT - 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 - 3.46iT - 67T^{2} \) |
| 71 | \( 1 + 3.46T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 10.3iT - 79T^{2} \) |
| 83 | \( 1 + 6.92T + 83T^{2} \) |
| 89 | \( 1 + 6iT - 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−10.47546214303283806740665857018, −9.466395746048916294117273105323, −8.944776749696174957110482465625, −7.949552815430817302605878953195, −6.90499212365134433398607276540, −5.81561721565226135772942722268, −4.92245804226977375307153500512, −4.81440769920302518278705001194, −2.80800676799281024732219574304, −1.34563628292917288796370845245,
0.35989702441147824434577908652, 2.31958156869378017543602943072, 3.64393078971262293095265687698, 4.65221579913749721114023753141, 5.65314879706759579258024135921, 6.84740644914127797473820899615, 7.12915560739169927688082980338, 7.914065693451782257353722101862, 9.829512095451914309134811929281, 10.02350780215104759168569466768