| L(s) = 1 | − 27·3-s − 114·5-s + 1.57e3·7-s + 729·9-s − 7.33e3·11-s − 3.80e3·13-s + 3.07e3·15-s − 6.60e3·17-s − 2.48e4·19-s − 4.25e4·21-s − 4.14e4·23-s − 6.51e4·25-s − 1.96e4·27-s − 4.16e4·29-s − 3.31e4·31-s + 1.97e5·33-s − 1.79e5·35-s − 3.64e4·37-s + 1.02e5·39-s − 6.39e5·41-s + 1.56e5·43-s − 8.31e4·45-s + 4.33e5·47-s + 1.66e6·49-s + 1.78e5·51-s + 7.86e5·53-s + 8.35e5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.407·5-s + 1.73·7-s + 1/3·9-s − 1.66·11-s − 0.479·13-s + 0.235·15-s − 0.326·17-s − 0.831·19-s − 1.00·21-s − 0.710·23-s − 0.833·25-s − 0.192·27-s − 0.316·29-s − 0.199·31-s + 0.958·33-s − 0.708·35-s − 0.118·37-s + 0.277·39-s − 1.44·41-s + 0.300·43-s − 0.135·45-s + 0.609·47-s + 2.01·49-s + 0.188·51-s + 0.725·53-s + 0.677·55-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 | \( 1 + 114 T + p^{7} T^{2} \) |
| 7 | \( 1 - 1576 T + p^{7} T^{2} \) |
| 11 | \( 1 + 7332 T + p^{7} T^{2} \) |
| 13 | \( 1 + 3802 T + p^{7} T^{2} \) |
| 17 | \( 1 + 6606 T + p^{7} T^{2} \) |
| 19 | \( 1 + 24860 T + p^{7} T^{2} \) |
| 23 | \( 1 + 41448 T + p^{7} T^{2} \) |
| 29 | \( 1 + 41610 T + p^{7} T^{2} \) |
| 31 | \( 1 + 33152 T + p^{7} T^{2} \) |
| 37 | \( 1 + 36466 T + p^{7} T^{2} \) |
| 41 | \( 1 + 639078 T + p^{7} T^{2} \) |
| 43 | \( 1 - 156412 T + p^{7} T^{2} \) |
| 47 | \( 1 - 433776 T + p^{7} T^{2} \) |
| 53 | \( 1 - 786078 T + p^{7} T^{2} \) |
| 59 | \( 1 + 745140 T + p^{7} T^{2} \) |
| 61 | \( 1 + 1660618 T + p^{7} T^{2} \) |
| 67 | \( 1 - 3290836 T + p^{7} T^{2} \) |
| 71 | \( 1 + 5716152 T + p^{7} T^{2} \) |
| 73 | \( 1 - 2659898 T + p^{7} T^{2} \) |
| 79 | \( 1 + 3807440 T + p^{7} T^{2} \) |
| 83 | \( 1 + 2229468 T + p^{7} T^{2} \) |
| 89 | \( 1 - 5991210 T + p^{7} T^{2} \) |
| 97 | \( 1 + 4060126 T + p^{7} T^{2} \) |
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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
−13.56500624692382763090710161063, −12.21931477724324608553848132058, −11.20268755172648883871783794747, −10.33812599850960390388893102661, −8.337264838598756159249573983743, −7.48293964396951728385229658052, −5.50115105699647054527637226538, −4.46507134142892990263090954210, −2.03777954935662729612290706522, 0,
2.03777954935662729612290706522, 4.46507134142892990263090954210, 5.50115105699647054527637226538, 7.48293964396951728385229658052, 8.337264838598756159249573983743, 10.33812599850960390388893102661, 11.20268755172648883871783794747, 12.21931477724324608553848132058, 13.56500624692382763090710161063
