| L(s) = 1 | + 84·3-s + 82·5-s − 456·7-s + 4.86e3·9-s + 2.52e3·11-s + 1.07e4·13-s + 6.88e3·15-s − 1.11e4·17-s − 4.12e3·19-s − 3.83e4·21-s + 8.17e4·23-s − 7.14e4·25-s + 2.25e5·27-s − 9.97e4·29-s − 4.04e4·31-s + 2.12e5·33-s − 3.73e4·35-s + 4.19e5·37-s + 9.05e5·39-s + 1.41e5·41-s + 6.90e5·43-s + 3.99e5·45-s − 6.82e5·47-s − 6.15e5·49-s − 9.36e5·51-s − 1.81e6·53-s + 2.06e5·55-s + ⋯ |
| L(s) = 1 | + 1.79·3-s + 0.293·5-s − 0.502·7-s + 2.22·9-s + 0.571·11-s + 1.36·13-s + 0.526·15-s − 0.550·17-s − 0.137·19-s − 0.902·21-s + 1.40·23-s − 0.913·25-s + 2.20·27-s − 0.759·29-s − 0.244·31-s + 1.02·33-s − 0.147·35-s + 1.36·37-s + 2.44·39-s + 0.320·41-s + 1.32·43-s + 0.653·45-s − 0.958·47-s − 0.747·49-s − 0.988·51-s − 1.67·53-s + 0.167·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.790520372\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.790520372\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 28 p T + p^{7} T^{2} \) |
| 5 | \( 1 - 82 T + p^{7} T^{2} \) |
| 7 | \( 1 + 456 T + p^{7} T^{2} \) |
| 11 | \( 1 - 2524 T + p^{7} T^{2} \) |
| 13 | \( 1 - 10778 T + p^{7} T^{2} \) |
| 17 | \( 1 + 11150 T + p^{7} T^{2} \) |
| 19 | \( 1 + 4124 T + p^{7} T^{2} \) |
| 23 | \( 1 - 81704 T + p^{7} T^{2} \) |
| 29 | \( 1 + 99798 T + p^{7} T^{2} \) |
| 31 | \( 1 + 40480 T + p^{7} T^{2} \) |
| 37 | \( 1 - 419442 T + p^{7} T^{2} \) |
| 41 | \( 1 - 141402 T + p^{7} T^{2} \) |
| 43 | \( 1 - 690428 T + p^{7} T^{2} \) |
| 47 | \( 1 + 682032 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1813118 T + p^{7} T^{2} \) |
| 59 | \( 1 - 966028 T + p^{7} T^{2} \) |
| 61 | \( 1 + 1887670 T + p^{7} T^{2} \) |
| 67 | \( 1 + 2965868 T + p^{7} T^{2} \) |
| 71 | \( 1 + 2548232 T + p^{7} T^{2} \) |
| 73 | \( 1 + 1680326 T + p^{7} T^{2} \) |
| 79 | \( 1 - 4038064 T + p^{7} T^{2} \) |
| 83 | \( 1 - 5385764 T + p^{7} T^{2} \) |
| 89 | \( 1 + 6473046 T + p^{7} T^{2} \) |
| 97 | \( 1 + 6065758 T + p^{7} T^{2} \) |
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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
−13.46408508332775722749647377858, −12.93386944122396361297540932404, −11.03553803175518738887548563317, −9.511119757754676092739872571953, −8.951120489062933395971176009476, −7.74552040124189339067595771093, −6.37047312434695373325764269714, −4.08613985239506777173045270964, −2.98483052247052051455788137478, −1.52391590118369923918176718330,
1.52391590118369923918176718330, 2.98483052247052051455788137478, 4.08613985239506777173045270964, 6.37047312434695373325764269714, 7.74552040124189339067595771093, 8.951120489062933395971176009476, 9.511119757754676092739872571953, 11.03553803175518738887548563317, 12.93386944122396361297540932404, 13.46408508332775722749647377858
