| L(s) = 1 | + 2-s + 3-s − 5-s + 6-s − 2·7-s + 8-s − 2·9-s − 10-s + 7·11-s − 13-s − 2·14-s − 15-s − 16-s − 4·17-s − 2·18-s − 14·19-s − 2·21-s + 7·22-s + 13·23-s + 24-s − 6·25-s − 26-s − 2·27-s + 2·29-s − 30-s + 6·31-s − 6·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 0.447·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s − 2/3·9-s − 0.316·10-s + 2.11·11-s − 0.277·13-s − 0.534·14-s − 0.258·15-s − 1/4·16-s − 0.970·17-s − 0.471·18-s − 3.21·19-s − 0.436·21-s + 1.49·22-s + 2.71·23-s + 0.204·24-s − 6/5·25-s − 0.196·26-s − 0.384·27-s + 0.371·29-s − 0.182·30-s + 1.07·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5329 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.187013242\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.187013242\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 73 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T + 7 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 - 7 T + 31 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + T + 23 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 13 T + 85 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 77 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + p T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 55 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 9 T + 139 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T + 115 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - T + 129 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 7 T + 97 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 5 T + 171 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.72040377265643180630712759419, −14.50350199733923038756179334344, −13.64832028481806944316760166387, −13.44628758855289881716339478914, −12.80964818245449182699867595010, −12.32243029588898619141806908808, −11.69217271527662664448533224314, −10.92521717080819984178439911224, −10.83068722341473055107727760735, −9.621328782484073169562313905267, −9.016243251200494593807958055995, −8.801704473564206926865340933121, −8.111173661069387653941008641711, −6.88525923069462626008397487116, −6.67511392041592030737607723858, −6.02831600838733131795377992512, −4.54572373547222112244451376293, −4.34665090980010480433814472594, −3.44857088236050752075861426439, −2.32362777020342723225021402452,
2.32362777020342723225021402452, 3.44857088236050752075861426439, 4.34665090980010480433814472594, 4.54572373547222112244451376293, 6.02831600838733131795377992512, 6.67511392041592030737607723858, 6.88525923069462626008397487116, 8.111173661069387653941008641711, 8.801704473564206926865340933121, 9.016243251200494593807958055995, 9.621328782484073169562313905267, 10.83068722341473055107727760735, 10.92521717080819984178439911224, 11.69217271527662664448533224314, 12.32243029588898619141806908808, 12.80964818245449182699867595010, 13.44628758855289881716339478914, 13.64832028481806944316760166387, 14.50350199733923038756179334344, 14.72040377265643180630712759419
