| L(s) = 1 | + 9·3-s − 64·5-s + 49·7-s + 81·9-s − 54·11-s + 738·13-s − 576·15-s − 848·17-s − 1.60e3·19-s + 441·21-s − 3.67e3·23-s + 971·25-s + 729·27-s − 4.33e3·29-s − 4.76e3·31-s − 486·33-s − 3.13e3·35-s − 2.09e3·37-s + 6.64e3·39-s − 6.11e3·41-s + 7.91e3·43-s − 5.18e3·45-s + 6.57e3·47-s + 2.40e3·49-s − 7.63e3·51-s − 7.89e3·53-s + 3.45e3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.14·5-s + 0.377·7-s + 1/3·9-s − 0.134·11-s + 1.21·13-s − 0.660·15-s − 0.711·17-s − 1.01·19-s + 0.218·21-s − 1.44·23-s + 0.310·25-s + 0.192·27-s − 0.956·29-s − 0.889·31-s − 0.0776·33-s − 0.432·35-s − 0.251·37-s + 0.699·39-s − 0.568·41-s + 0.652·43-s − 0.381·45-s + 0.433·47-s + 1/7·49-s − 0.410·51-s − 0.386·53-s + 0.154·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{2} T \) |
| 7 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 + 64 T + p^{5} T^{2} \) |
| 11 | \( 1 + 54 T + p^{5} T^{2} \) |
| 13 | \( 1 - 738 T + p^{5} T^{2} \) |
| 17 | \( 1 + 848 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1604 T + p^{5} T^{2} \) |
| 23 | \( 1 + 3670 T + p^{5} T^{2} \) |
| 29 | \( 1 + 4330 T + p^{5} T^{2} \) |
| 31 | \( 1 + 4760 T + p^{5} T^{2} \) |
| 37 | \( 1 + 2094 T + p^{5} T^{2} \) |
| 41 | \( 1 + 6116 T + p^{5} T^{2} \) |
| 43 | \( 1 - 7916 T + p^{5} T^{2} \) |
| 47 | \( 1 - 6572 T + p^{5} T^{2} \) |
| 53 | \( 1 + 7894 T + p^{5} T^{2} \) |
| 59 | \( 1 + 41664 T + p^{5} T^{2} \) |
| 61 | \( 1 + 26570 T + p^{5} T^{2} \) |
| 67 | \( 1 + 41736 T + p^{5} T^{2} \) |
| 71 | \( 1 - 83574 T + p^{5} T^{2} \) |
| 73 | \( 1 + 42314 T + p^{5} T^{2} \) |
| 79 | \( 1 - 508 T + p^{5} T^{2} \) |
| 83 | \( 1 + 8364 T + p^{5} T^{2} \) |
| 89 | \( 1 + 49220 T + p^{5} T^{2} \) |
| 97 | \( 1 - 159670 T + p^{5} T^{2} \) |
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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
−11.33088169358493994842137367107, −10.59515833574378033149620756321, −9.071916547299560489892329086060, −8.261965650540978813068924289279, −7.47586481468669741687775750765, −6.10073025802046248209309105820, −4.37412813733597309839785724743, −3.58637845354408141371162686901, −1.87748905567315023976649407164, 0,
1.87748905567315023976649407164, 3.58637845354408141371162686901, 4.37412813733597309839785724743, 6.10073025802046248209309105820, 7.47586481468669741687775750765, 8.261965650540978813068924289279, 9.071916547299560489892329086060, 10.59515833574378033149620756321, 11.33088169358493994842137367107
