| L(s) = 1 | − 27·3-s + 270·5-s − 1.11e3·7-s + 729·9-s + 5.72e3·11-s − 4.57e3·13-s − 7.29e3·15-s − 3.65e4·17-s − 5.17e4·19-s + 3.00e4·21-s − 2.22e4·23-s − 5.22e3·25-s − 1.96e4·27-s − 1.57e5·29-s + 1.03e5·31-s − 1.54e5·33-s − 3.00e5·35-s − 9.48e4·37-s + 1.23e5·39-s + 6.59e5·41-s + 7.57e4·43-s + 1.96e5·45-s − 4.05e5·47-s + 4.13e5·49-s + 9.87e5·51-s − 1.34e6·53-s + 1.54e6·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.965·5-s − 1.22·7-s + 1/3·9-s + 1.29·11-s − 0.576·13-s − 0.557·15-s − 1.80·17-s − 1.73·19-s + 0.707·21-s − 0.381·23-s − 0.0668·25-s − 0.192·27-s − 1.19·29-s + 0.626·31-s − 0.748·33-s − 1.18·35-s − 0.307·37-s + 0.333·39-s + 1.49·41-s + 0.145·43-s + 0.321·45-s − 0.569·47-s + 0.501·49-s + 1.04·51-s − 1.24·53-s + 1.25·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{3} T \) |
| good | 5 | \( 1 - 54 p T + p^{7} T^{2} \) |
| 7 | \( 1 + 1112 T + p^{7} T^{2} \) |
| 11 | \( 1 - 5724 T + p^{7} T^{2} \) |
| 13 | \( 1 + 4570 T + p^{7} T^{2} \) |
| 17 | \( 1 + 36558 T + p^{7} T^{2} \) |
| 19 | \( 1 + 51740 T + p^{7} T^{2} \) |
| 23 | \( 1 + 22248 T + p^{7} T^{2} \) |
| 29 | \( 1 + 157194 T + p^{7} T^{2} \) |
| 31 | \( 1 - 103936 T + p^{7} T^{2} \) |
| 37 | \( 1 + 94834 T + p^{7} T^{2} \) |
| 41 | \( 1 - 659610 T + p^{7} T^{2} \) |
| 43 | \( 1 - 75772 T + p^{7} T^{2} \) |
| 47 | \( 1 + 405648 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1346274 T + p^{7} T^{2} \) |
| 59 | \( 1 - 1303884 T + p^{7} T^{2} \) |
| 61 | \( 1 - 30062 p T + p^{7} T^{2} \) |
| 67 | \( 1 + 1369388 T + p^{7} T^{2} \) |
| 71 | \( 1 + 2714040 T + p^{7} T^{2} \) |
| 73 | \( 1 - 2868794 T + p^{7} T^{2} \) |
| 79 | \( 1 - 1129648 T + p^{7} T^{2} \) |
| 83 | \( 1 + 5912028 T + p^{7} T^{2} \) |
| 89 | \( 1 + 897750 T + p^{7} T^{2} \) |
| 97 | \( 1 - 13719074 T + p^{7} T^{2} \) |
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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
−13.36523989078244672715467420737, −12.59395830261224074868032566621, −11.16785668576431430521990292433, −9.882269363854955837609103233618, −9.031593177910491907801163682837, −6.71224496459768726828050010923, −6.10727173926419474050698802730, −4.21245170640410358770610947569, −2.10008144971831805440499734162, 0,
2.10008144971831805440499734162, 4.21245170640410358770610947569, 6.10727173926419474050698802730, 6.71224496459768726828050010923, 9.031593177910491907801163682837, 9.882269363854955837609103233618, 11.16785668576431430521990292433, 12.59395830261224074868032566621, 13.36523989078244672715467420737
