| L(s) = 1 | − 2·2-s + 7-s + 4·8-s + 3·11-s − 13-s − 2·14-s − 4·16-s + 3·17-s + 6·19-s − 6·22-s + 2·23-s − 4·25-s + 2·26-s + 29-s + 3·31-s − 6·34-s + 37-s − 12·38-s − 9·41-s + 3·43-s − 4·46-s − 47-s − 2·49-s + 8·50-s − 5·53-s + 4·56-s − 2·58-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.377·7-s + 1.41·8-s + 0.904·11-s − 0.277·13-s − 0.534·14-s − 16-s + 0.727·17-s + 1.37·19-s − 1.27·22-s + 0.417·23-s − 4/5·25-s + 0.392·26-s + 0.185·29-s + 0.538·31-s − 1.02·34-s + 0.164·37-s − 1.94·38-s − 1.40·41-s + 0.457·43-s − 0.589·46-s − 0.145·47-s − 2/7·49-s + 1.13·50-s − 0.686·53-s + 0.534·56-s − 0.262·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4293 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4293 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3753989102\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3753989102\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 6 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 3 T + 19 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 - T + 13 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 - T - 27 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 + T - 53 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + T + 103 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 96 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 108 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 + T - 17 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 15 T + 115 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
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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
−17.8276501718, −17.2144502412, −16.9720195645, −16.5133570096, −15.7773721955, −15.2803937855, −14.4844995664, −14.1115510914, −13.6339017814, −13.0502738968, −12.0787017690, −11.8554825113, −11.1295100884, −10.3329183570, −9.86387749619, −9.39491409149, −8.93547444646, −8.20838846635, −7.74233833483, −7.11434964153, −6.11727954681, −5.15974543666, −4.41883882723, −3.29018322607, −1.35966579559,
1.35966579559, 3.29018322607, 4.41883882723, 5.15974543666, 6.11727954681, 7.11434964153, 7.74233833483, 8.20838846635, 8.93547444646, 9.39491409149, 9.86387749619, 10.3329183570, 11.1295100884, 11.8554825113, 12.0787017690, 13.0502738968, 13.6339017814, 14.1115510914, 14.4844995664, 15.2803937855, 15.7773721955, 16.5133570096, 16.9720195645, 17.2144502412, 17.8276501718
