| L(s) = 1 | + 2·2-s + 2·4-s − 2·5-s − 7-s − 4·10-s − 11-s − 2·14-s − 4·16-s − 4·17-s − 7·19-s − 4·20-s − 2·22-s − 25-s − 2·28-s + 4·29-s − 4·31-s − 8·32-s − 8·34-s + 2·35-s − 4·37-s − 14·38-s − 9·41-s − 6·43-s − 2·44-s − 3·47-s + 49-s − 2·50-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 0.894·5-s − 0.377·7-s − 1.26·10-s − 0.301·11-s − 0.534·14-s − 16-s − 0.970·17-s − 1.60·19-s − 0.894·20-s − 0.426·22-s − 1/5·25-s − 0.377·28-s + 0.742·29-s − 0.718·31-s − 1.41·32-s − 1.37·34-s + 0.338·35-s − 0.657·37-s − 2.27·38-s − 1.40·41-s − 0.914·43-s − 0.301·44-s − 0.437·47-s + 1/7·49-s − 0.282·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33327 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33327 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7787750725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7787750725\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.93756018634585, −14.58074698520131, −13.88869049507254, −13.26455254141561, −13.03650108836701, −12.51917512485192, −11.86173579676026, −11.61169099247029, −10.90486858931646, −10.45604671687965, −9.747634062517209, −8.846262560485219, −8.586278489316317, −7.878445158907065, −7.121968585513824, −6.508681593934245, −6.296625069975197, −5.368848983801890, −4.780381287384947, −4.376782021120012, −3.634464633864103, −3.340976771011779, −2.415227876952615, −1.841439046886328, −0.2418700960055086,
0.2418700960055086, 1.841439046886328, 2.415227876952615, 3.340976771011779, 3.634464633864103, 4.376782021120012, 4.780381287384947, 5.368848983801890, 6.296625069975197, 6.508681593934245, 7.121968585513824, 7.878445158907065, 8.586278489316317, 8.846262560485219, 9.747634062517209, 10.45604671687965, 10.90486858931646, 11.61169099247029, 11.86173579676026, 12.51917512485192, 13.03650108836701, 13.26455254141561, 13.88869049507254, 14.58074698520131, 14.93756018634585
