L(s) = 1 | + 5.65·2-s + 15.5·3-s + 32.0·4-s − 227. i·5-s + 88.1·6-s + 132. i·7-s + 181.·8-s + 243·9-s − 1.28e3i·10-s − 1.00e3i·11-s + 498.·12-s − 597.·13-s + 746. i·14-s − 3.55e3i·15-s + 1.02e3·16-s + 500. i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.500·4-s − 1.82i·5-s + 0.408·6-s + 0.384i·7-s + 0.353·8-s + 0.333·9-s − 1.28i·10-s − 0.758i·11-s + 0.288·12-s − 0.272·13-s + 0.272i·14-s − 1.05i·15-s + 0.250·16-s + 0.101i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(3.448924008\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.448924008\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 5.65T \) |
| 3 | \( 1 - 15.5T \) |
| 23 | \( 1 + (-2.71e3 + 1.18e4i)T \) |
good | 5 | \( 1 + 227. iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 132. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 1.00e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 597.T + 4.82e6T^{2} \) |
| 17 | \( 1 - 500. iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 2.11e3iT - 4.70e7T^{2} \) |
| 29 | \( 1 + 1.23e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + 3.93e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 2.64e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 1.01e5T + 4.75e9T^{2} \) |
| 43 | \( 1 - 1.85e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 1.41e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + 1.51e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 2.26e5T + 4.21e10T^{2} \) |
| 61 | \( 1 - 1.69e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 4.17e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 1.02e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 5.48e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 4.12e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 2.31e4iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 7.79e4iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 4.61e5iT - 8.32e11T^{2} \) |
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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
−12.22241877248848035169977813229, −10.93398941453446769362847575768, −9.322836903248007998058307615894, −8.709083114741486821449754158607, −7.62226388426388682150827539637, −5.90035668106582108716621658984, −4.93818059470010621055447524778, −3.85512485196779449912833831787, −2.21732442091567249799961453670, −0.75152702384116268516758137666,
2.02187027595595933896836343740, 3.10674569372893818750711868526, 4.10026972661832029934829900785, 5.84801439779706404076753501436, 7.18400242670644634798862035005, 7.49109027996835726442358948840, 9.480262712624057811262577919398, 10.46978300187529795387590016966, 11.23640248515836762641482046264, 12.44811317075540734861341861684