| L(s) = 1 | + 2-s + 3-s + 3·5-s + 6-s − 2·7-s − 8-s + 3·10-s + 8·11-s − 3·13-s − 2·14-s + 3·15-s − 16-s + 2·17-s − 19-s − 2·21-s + 8·22-s − 23-s − 24-s + 5·25-s − 3·26-s − 27-s + 3·30-s − 12·31-s + 8·33-s + 2·34-s − 6·35-s + 8·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1.34·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 0.948·10-s + 2.41·11-s − 0.832·13-s − 0.534·14-s + 0.774·15-s − 1/4·16-s + 0.485·17-s − 0.229·19-s − 0.436·21-s + 1.70·22-s − 0.208·23-s − 0.204·24-s + 25-s − 0.588·26-s − 0.192·27-s + 0.547·30-s − 2.15·31-s + 1.39·33-s + 0.342·34-s − 1.01·35-s + 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 636804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 636804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.398622336\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.398622336\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 3 T - 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 4 T - 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 6 T - 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 13 T + 110 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 7 T - 22 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 4 T - 73 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−10.56970348771225715400306042465, −9.731010105868434947266154590066, −9.391327515810790811710602612951, −9.299285308695141267304690956088, −9.263406870765290868476601362105, −8.460384991077613465083966317582, −7.914812981667487845865926097214, −7.36645604751809389312798593152, −6.77916338915524917530345572932, −6.51561105484782949040035350463, −6.15857133201505001567874599934, −5.50263817137417872688784373998, −5.38835630499415343583641879063, −4.38907951136911469032006443717, −4.18752185103194328842732811412, −3.33591707128064119574077091548, −3.31556961273887671445097648312, −2.19276565882406798027273996228, −1.98113746655403412144014082232, −0.950263605815324809574173240054,
0.950263605815324809574173240054, 1.98113746655403412144014082232, 2.19276565882406798027273996228, 3.31556961273887671445097648312, 3.33591707128064119574077091548, 4.18752185103194328842732811412, 4.38907951136911469032006443717, 5.38835630499415343583641879063, 5.50263817137417872688784373998, 6.15857133201505001567874599934, 6.51561105484782949040035350463, 6.77916338915524917530345572932, 7.36645604751809389312798593152, 7.914812981667487845865926097214, 8.460384991077613465083966317582, 9.263406870765290868476601362105, 9.299285308695141267304690956088, 9.391327515810790811710602612951, 9.731010105868434947266154590066, 10.56970348771225715400306042465
