| L(s) = 1 | + 2-s + 3-s + 4-s − 2.97·5-s + 6-s + 4.16·7-s + 8-s + 9-s − 2.97·10-s − 4.19·11-s + 12-s + 0.854·13-s + 4.16·14-s − 2.97·15-s + 16-s + 17-s + 18-s − 5.91·19-s − 2.97·20-s + 4.16·21-s − 4.19·22-s − 4.66·23-s + 24-s + 3.84·25-s + 0.854·26-s + 27-s + 4.16·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.32·5-s + 0.408·6-s + 1.57·7-s + 0.353·8-s + 0.333·9-s − 0.940·10-s − 1.26·11-s + 0.288·12-s + 0.236·13-s + 1.11·14-s − 0.767·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s − 1.35·19-s − 0.664·20-s + 0.909·21-s − 0.894·22-s − 0.971·23-s + 0.204·24-s + 0.768·25-s + 0.167·26-s + 0.192·27-s + 0.787·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 5 | \( 1 + 2.97T + 5T^{2} \) |
| 7 | \( 1 - 4.16T + 7T^{2} \) |
| 11 | \( 1 + 4.19T + 11T^{2} \) |
| 13 | \( 1 - 0.854T + 13T^{2} \) |
| 19 | \( 1 + 5.91T + 19T^{2} \) |
| 23 | \( 1 + 4.66T + 23T^{2} \) |
| 29 | \( 1 + 6.51T + 29T^{2} \) |
| 31 | \( 1 + 6.77T + 31T^{2} \) |
| 37 | \( 1 - 4.01T + 37T^{2} \) |
| 41 | \( 1 + 5.28T + 41T^{2} \) |
| 43 | \( 1 + 8.96T + 43T^{2} \) |
| 47 | \( 1 + 1.57T + 47T^{2} \) |
| 53 | \( 1 + 8.73T + 53T^{2} \) |
| 61 | \( 1 - 1.25T + 61T^{2} \) |
| 67 | \( 1 - 1.32T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 + 2.48T + 73T^{2} \) |
| 79 | \( 1 - 6.76T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 + 1.90T + 89T^{2} \) |
| 97 | \( 1 + 4.72T + 97T^{2} \) |
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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
−7.83875294697547243361352123352, −7.30780614318514208626520478748, −6.27401060243578610577351227258, −5.23848810672064094447125095386, −4.79511467613304613003915560513, −3.96823751929176007284539680711, −3.51643932945228327393356079561, −2.33424594548734319667313268742, −1.69951536726511065828997682897, 0,
1.69951536726511065828997682897, 2.33424594548734319667313268742, 3.51643932945228327393356079561, 3.96823751929176007284539680711, 4.79511467613304613003915560513, 5.23848810672064094447125095386, 6.27401060243578610577351227258, 7.30780614318514208626520478748, 7.83875294697547243361352123352
