| L(s) = 1 | + 23·4-s − 38·7-s + 604·13-s + 273·16-s − 608·19-s + 161·25-s − 874·28-s + 478·31-s + 1.48e3·37-s − 1.96e3·43-s − 3.71e3·49-s + 1.38e4·52-s − 632·61-s + 391·64-s + 9.24e3·67-s − 6.06e3·73-s − 1.39e4·76-s − 2.09e4·79-s − 2.29e4·91-s − 1.30e4·97-s + 3.70e3·100-s − 1.53e4·103-s + 4.64e3·109-s − 1.03e4·112-s + 1.41e4·121-s + 1.09e4·124-s + 127-s + ⋯ |
| L(s) = 1 | + 1.43·4-s − 0.775·7-s + 3.57·13-s + 1.06·16-s − 1.68·19-s + 0.257·25-s − 1.11·28-s + 0.497·31-s + 1.08·37-s − 1.06·43-s − 1.54·49-s + 5.13·52-s − 0.169·61-s + 0.0954·64-s + 2.05·67-s − 1.13·73-s − 2.42·76-s − 3.34·79-s − 2.77·91-s − 1.38·97-s + 0.370·100-s − 1.44·103-s + 0.391·109-s − 0.827·112-s + 0.966·121-s + 0.715·124-s + 6.20e−5·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.239978327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.239978327\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 23 T^{2} + p^{8} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 161 T^{2} + p^{8} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 19 T + p^{4} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 14153 T^{2} + p^{8} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 302 T + p^{4} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 4354 T^{2} + p^{8} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 16 p T + p^{4} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 469682 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 954878 T^{2} + p^{8} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 239 T + p^{4} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 20 p T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 5599538 T^{2} + p^{8} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 982 T + p^{4} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 5067806 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 13243313 T^{2} + p^{8} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 15696638 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 316 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4622 T + p^{4} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 47518238 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 3031 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10450 T + p^{4} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 64676047 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 76456478 T^{2} + p^{8} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6517 T + p^{4} T^{2} )^{2} \) |
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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
−16.50259104642843069262332811953, −16.23279407038076774837417964119, −15.76107984422775803842044476115, −15.31650150181296386185682261607, −14.57240935625533123466617305303, −13.57193457608714596607695177070, −13.11924741795611269714341046009, −12.58931403462131941534971713157, −11.42635878253882303189727707945, −11.18386426520262237837112012986, −10.66317134386680375165190413963, −9.810941832294436441718305423684, −8.537052936300178784252267315297, −8.356504099295465831842273148100, −6.92663286302957008782946200185, −6.25477143298326408900515905722, −6.04368205348361137920981572860, −4.03088518790966787447861190872, −3.05553397015761697482722952315, −1.50186504539961395047944735848,
1.50186504539961395047944735848, 3.05553397015761697482722952315, 4.03088518790966787447861190872, 6.04368205348361137920981572860, 6.25477143298326408900515905722, 6.92663286302957008782946200185, 8.356504099295465831842273148100, 8.537052936300178784252267315297, 9.810941832294436441718305423684, 10.66317134386680375165190413963, 11.18386426520262237837112012986, 11.42635878253882303189727707945, 12.58931403462131941534971713157, 13.11924741795611269714341046009, 13.57193457608714596607695177070, 14.57240935625533123466617305303, 15.31650150181296386185682261607, 15.76107984422775803842044476115, 16.23279407038076774837417964119, 16.50259104642843069262332811953
