L(s) = 1 | − 2.06i·2-s − 1.78i·3-s − 2.25·4-s − 0.290i·5-s − 3.68·6-s + (2.64 − 0.129i)7-s + 0.517i·8-s − 0.200·9-s − 0.598·10-s + 4.02i·12-s − 2.85·13-s + (−0.267 − 5.44i)14-s − 0.519·15-s − 3.43·16-s − 6.77·17-s + 0.413i·18-s + ⋯ |
L(s) = 1 | − 1.45i·2-s − 1.03i·3-s − 1.12·4-s − 0.129i·5-s − 1.50·6-s + (0.998 − 0.0491i)7-s + 0.183i·8-s − 0.0668·9-s − 0.189·10-s + 1.16i·12-s − 0.792·13-s + (−0.0716 − 1.45i)14-s − 0.134·15-s − 0.858·16-s − 1.64·17-s + 0.0975i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 - 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.563 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.625667 + 1.18411i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.625667 + 1.18411i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-2.64 + 0.129i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 2.06iT - 2T^{2} \) |
| 3 | \( 1 + 1.78iT - 3T^{2} \) |
| 5 | \( 1 + 0.290iT - 5T^{2} \) |
| 13 | \( 1 + 2.85T + 13T^{2} \) |
| 17 | \( 1 + 6.77T + 17T^{2} \) |
| 19 | \( 1 + 2.49T + 19T^{2} \) |
| 23 | \( 1 + 4.08T + 23T^{2} \) |
| 29 | \( 1 + 6.91iT - 29T^{2} \) |
| 31 | \( 1 + 5.12iT - 31T^{2} \) |
| 37 | \( 1 - 6.31T + 37T^{2} \) |
| 41 | \( 1 + 9.51T + 41T^{2} \) |
| 43 | \( 1 + 2.51iT - 43T^{2} \) |
| 47 | \( 1 - 7.65iT - 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 - 4.12iT - 59T^{2} \) |
| 61 | \( 1 - 6.06T + 61T^{2} \) |
| 67 | \( 1 - 2.66T + 67T^{2} \) |
| 71 | \( 1 - 5.53T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + 13.9iT - 79T^{2} \) |
| 83 | \( 1 - 0.375T + 83T^{2} \) |
| 89 | \( 1 + 13.8iT - 89T^{2} \) |
| 97 | \( 1 + 5.85iT - 97T^{2} \) |
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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
−9.887678033390361239429021961087, −8.883658574562481942456235309205, −8.051204725110290569894793720210, −7.14531147972975068567339583919, −6.25988522589761779569366444189, −4.72200610195602199316455156200, −4.12236231751854803261730634241, −2.37070622139410203045444688495, −2.00964454852610727201123828375, −0.63468769728626880953916169641,
2.22472132535430933846829870263, 3.98733970598967443820066096368, 4.87407368499258826072446762430, 5.22153186130394502818980321189, 6.64428316202179545603435546338, 7.11764149004883324173238343461, 8.334880087297304729130805352357, 8.726857299419914857976360571744, 9.708423918634411456975335163574, 10.69385238487210636865159741466