L(s) = 1 | + (46.0 + 8.05i)3-s + (565. − 565. i)7-s + (2.05e3 + 741. i)9-s + 2.36e3i·11-s + (−732. − 732. i)13-s + (−2.28e4 − 2.28e4i)17-s − 3.87e4i·19-s + (3.05e4 − 2.14e4i)21-s + (2.39e4 − 2.39e4i)23-s + (8.88e4 + 5.07e4i)27-s + 1.72e5·29-s − 1.19e5·31-s + (−1.90e4 + 1.08e5i)33-s + (−9.36e4 + 9.36e4i)37-s + (−2.78e4 − 3.96e4i)39-s + ⋯ |
L(s) = 1 | + (0.985 + 0.172i)3-s + (0.622 − 0.622i)7-s + (0.940 + 0.339i)9-s + 0.534i·11-s + (−0.0924 − 0.0924i)13-s + (−1.13 − 1.13i)17-s − 1.29i·19-s + (0.720 − 0.506i)21-s + (0.409 − 0.409i)23-s + (0.868 + 0.496i)27-s + 1.31·29-s − 0.717·31-s + (−0.0921 + 0.526i)33-s + (−0.304 + 0.304i)37-s + (−0.0751 − 0.106i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(3.144461328\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.144461328\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-46.0 - 8.05i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-565. + 565. i)T - 8.23e5iT^{2} \) |
| 11 | \( 1 - 2.36e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 + (732. + 732. i)T + 6.27e7iT^{2} \) |
| 17 | \( 1 + (2.28e4 + 2.28e4i)T + 4.10e8iT^{2} \) |
| 19 | \( 1 + 3.87e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + (-2.39e4 + 2.39e4i)T - 3.40e9iT^{2} \) |
| 29 | \( 1 - 1.72e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.19e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (9.36e4 - 9.36e4i)T - 9.49e10iT^{2} \) |
| 41 | \( 1 + 6.73e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 + (2.65e5 + 2.65e5i)T + 2.71e11iT^{2} \) |
| 47 | \( 1 + (-2.18e5 - 2.18e5i)T + 5.06e11iT^{2} \) |
| 53 | \( 1 + (-9.79e3 + 9.79e3i)T - 1.17e12iT^{2} \) |
| 59 | \( 1 - 1.54e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.94e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + (8.98e5 - 8.98e5i)T - 6.06e12iT^{2} \) |
| 71 | \( 1 + 2.33e6iT - 9.09e12T^{2} \) |
| 73 | \( 1 + (2.64e6 + 2.64e6i)T + 1.10e13iT^{2} \) |
| 79 | \( 1 + 3.05e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + (-2.34e6 + 2.34e6i)T - 2.71e13iT^{2} \) |
| 89 | \( 1 + 3.61e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-3.89e6 + 3.89e6i)T - 8.07e13iT^{2} \) |
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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
−10.32400069166773290330204727632, −9.224062483792112257580492467415, −8.596993649578735477006211249013, −7.37668006509629847891103297972, −6.88572269038259260416737827646, −4.92525495266930074686751277043, −4.35005633548813039487484472367, −2.94895693055532671379660286994, −1.98256788533433913607024239800, −0.58960208945511585640683955098,
1.32222310641343908989677116673, 2.22478261057933853117071684318, 3.42517062243281475223281718097, 4.49997096898908765140643356697, 5.84321899010880265293555890859, 6.92545239171561168590683140341, 8.255309301246566083506235398832, 8.481699884808899373402466622572, 9.617030084122462368234877438278, 10.63089300645562734376936637788