| L(s) = 1 | − 77.5i·3-s + (184. + 210. i)5-s + 1.38e3i·7-s − 3.82e3·9-s + 414.·11-s + 1.02e4i·13-s + (1.63e4 − 1.42e4i)15-s − 1.43e4i·17-s + 3.69e4·19-s + 1.07e5·21-s − 1.43e4i·23-s + (−1.03e4 + 7.74e4i)25-s + 1.26e5i·27-s − 1.48e5·29-s − 2.52e5·31-s + ⋯ |
| L(s) = 1 | − 1.65i·3-s + (0.658 + 0.752i)5-s + 1.52i·7-s − 1.74·9-s + 0.0938·11-s + 1.29i·13-s + (1.24 − 1.09i)15-s − 0.708i·17-s + 1.23·19-s + 2.52·21-s − 0.246i·23-s + (−0.132 + 0.991i)25-s + 1.23i·27-s − 1.13·29-s − 1.52·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.5855831474\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5855831474\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-184. - 210. i)T \) |
| good | 3 | \( 1 + 77.5iT - 2.18e3T^{2} \) |
| 7 | \( 1 - 1.38e3iT - 8.23e5T^{2} \) |
| 11 | \( 1 - 414.T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.02e4iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 1.43e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 - 3.69e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.43e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 + 1.48e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.52e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.78e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 6.20e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.43e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 1.14e6iT - 5.06e11T^{2} \) |
| 53 | \( 1 + 1.17e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 4.67e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.52e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.74e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 1.14e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.06e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 + 7.10e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 3.67e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 4.61e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.04e7iT - 8.07e13T^{2} \) |
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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
−11.21252241612284867234806528865, −9.465572473268366857791993919667, −8.984073051921518062905471945291, −7.63012322692609858631415822134, −6.93144458547480578904194273747, −6.06949349741313700579565939475, −5.34111056835594237403919483114, −3.14403639285051067023097198695, −2.15484537490979679156319608059, −1.60906646464501357392669166629,
0.11772446578275625443962706435, 1.35712481634240554589117510462, 3.30258141606966032291187112298, 3.99370204169020414913973575718, 5.07358616678507026498107669815, 5.68807544731882065945735558542, 7.32087061902933077800278704227, 8.446789257106525749264255445729, 9.418027998707120069050686862941, 10.22190524635228875977044941529
