L(s) = 1 | + 10.6·2-s + 49.0·4-s + 66.6i·5-s + 306. i·7-s − 159.·8-s + 708. i·10-s + 998. i·11-s − 1.93e3·13-s + 3.26e3i·14-s − 4.83e3·16-s − 9.46e3i·17-s − 9.74e3i·19-s + 3.26e3i·20-s + 1.06e4i·22-s + (−1.20e4 + 1.83e3i)23-s + ⋯ |
L(s) = 1 | + 1.32·2-s + 0.765·4-s + 0.532i·5-s + 0.894i·7-s − 0.311·8-s + 0.708i·10-s + 0.750i·11-s − 0.878·13-s + 1.18i·14-s − 1.17·16-s − 1.92i·17-s − 1.42i·19-s + 0.408i·20-s + 0.997i·22-s + (−0.988 + 0.150i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.150i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.988 + 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.6158807347\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6158807347\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 + (1.20e4 - 1.83e3i)T \) |
good | 2 | \( 1 - 10.6T + 64T^{2} \) |
| 5 | \( 1 - 66.6iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 306. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 998. iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 1.93e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 9.46e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 9.74e3iT - 4.70e7T^{2} \) |
| 29 | \( 1 + 4.41e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 1.29e4T + 8.87e8T^{2} \) |
| 37 | \( 1 - 9.38e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 3.57e4T + 4.75e9T^{2} \) |
| 43 | \( 1 - 674. iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 1.02e5T + 1.07e10T^{2} \) |
| 53 | \( 1 - 1.85e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 7.62e4T + 4.21e10T^{2} \) |
| 61 | \( 1 + 4.05e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 4.14e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 1.12e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + 1.20e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 3.85e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 1.30e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 1.03e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 2.70e4iT - 8.32e11T^{2} \) |
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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
−11.87332251732036313013442016185, −11.42867537975123389264709955457, −9.816860125753993741265045574349, −9.050388349765837496514590443725, −7.38256748812814738212375013156, −6.55114467783345130964598298507, −5.24649000324945631268700236840, −4.65348057163564662477317496938, −3.04430983608046329618963431444, −2.35437014179445229706352223695,
0.093706332985268217177576707580, 1.81870763986128988856372586624, 3.58723657166302702769523888099, 4.16516456733737986741594849286, 5.45408284736400596682792737915, 6.23326996852457525087215121916, 7.63089236254998770374869380016, 8.696200116339738576151081083863, 10.03802285903317574009185673856, 10.99335912566390422510783010836