| L(s) = 1 | − 3-s − 2·7-s + 4·11-s + 3·13-s − 4·17-s − 8·19-s + 2·21-s + 4·23-s + 5·25-s + 27-s + 6·31-s − 4·33-s − 10·37-s − 3·39-s − 4·41-s − 9·43-s + 10·47-s − 11·49-s + 4·51-s + 4·53-s + 8·57-s − 14·59-s − 11·61-s + 3·67-s − 4·69-s + 14·71-s + 11·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1.20·11-s + 0.832·13-s − 0.970·17-s − 1.83·19-s + 0.436·21-s + 0.834·23-s + 25-s + 0.192·27-s + 1.07·31-s − 0.696·33-s − 1.64·37-s − 0.480·39-s − 0.624·41-s − 1.37·43-s + 1.45·47-s − 1.57·49-s + 0.560·51-s + 0.549·53-s + 1.05·57-s − 1.82·59-s − 1.40·61-s + 0.366·67-s − 0.481·69-s + 1.66·71-s + 1.28·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.157124780\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.157124780\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 19 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 4 T - 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 9 T + 38 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 10 T + 53 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 14 T + 137 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 14 T + 125 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 14 T + 107 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−10.47140252490331208747956144629, −9.993617443608869853442800013521, −9.296795405918880048668634561667, −9.021324787538220114908623211405, −8.776645008710452617809812922838, −8.329606155249876670375813681076, −7.87839890661223125054854045295, −7.01332594763960408665411084164, −6.61165737228390701623702765014, −6.57377424807135374838341145114, −6.25237145859147901907165990483, −5.57814408467689425737300996369, −4.94932860178085626855492071007, −4.53488921649565395276253547459, −4.11601781371529114005569547984, −3.34551188670806943062898976178, −3.14962463111515055093167669483, −2.13091480285622525382509842280, −1.54886931932192174762563354619, −0.54194631215494794124172825010,
0.54194631215494794124172825010, 1.54886931932192174762563354619, 2.13091480285622525382509842280, 3.14962463111515055093167669483, 3.34551188670806943062898976178, 4.11601781371529114005569547984, 4.53488921649565395276253547459, 4.94932860178085626855492071007, 5.57814408467689425737300996369, 6.25237145859147901907165990483, 6.57377424807135374838341145114, 6.61165737228390701623702765014, 7.01332594763960408665411084164, 7.87839890661223125054854045295, 8.329606155249876670375813681076, 8.776645008710452617809812922838, 9.021324787538220114908623211405, 9.296795405918880048668634561667, 9.993617443608869853442800013521, 10.47140252490331208747956144629
