| L(s) = 1 | − 2·3-s + 3·5-s − 2·7-s − 3·9-s − 6·15-s + 3·17-s + 16·19-s + 4·21-s + 3·23-s − 2·25-s + 14·27-s + 5·29-s − 6·35-s + 7·41-s + 12·43-s − 9·45-s − 9·47-s − 9·49-s − 6·51-s − 17·53-s − 32·57-s − 9·59-s + 2·61-s + 6·63-s − 18·67-s − 6·69-s − 19·71-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.34·5-s − 0.755·7-s − 9-s − 1.54·15-s + 0.727·17-s + 3.67·19-s + 0.872·21-s + 0.625·23-s − 2/5·25-s + 2.69·27-s + 0.928·29-s − 1.01·35-s + 1.09·41-s + 1.82·43-s − 1.34·45-s − 1.31·47-s − 9/7·49-s − 0.840·51-s − 2.33·53-s − 4.23·57-s − 1.17·59-s + 0.256·61-s + 0.755·63-s − 2.19·67-s − 0.722·69-s − 2.25·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.065207317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.065207317\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 3 T + 11 T^{2} - 24 T^{3} + 54 T^{4} - 24 p T^{5} + 11 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^3$ | \( 1 + 15 T^{2} + 56 T^{4} + 15 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 3 T + 35 T^{2} - 96 T^{3} + 786 T^{4} - 96 p T^{5} + 35 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 3 T - 19 T^{2} + 66 T^{3} + 24 T^{4} + 66 p T^{5} - 19 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 5 T - 31 T^{2} + 10 T^{3} + 1570 T^{4} + 10 p T^{5} - 31 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 45 T^{2} + 2024 T^{4} - 45 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 39 T^{2} + 2120 T^{4} + 39 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 7 T - 37 T^{2} - 28 T^{3} + 3706 T^{4} - 28 p T^{5} - 37 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 12 T + 55 T^{2} - 36 T^{3} - 120 T^{4} - 36 p T^{5} + 55 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 9 T + 125 T^{2} + 882 T^{3} + 8664 T^{4} + 882 p T^{5} + 125 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 17 T + 119 T^{2} + 1088 T^{3} + 10774 T^{4} + 1088 p T^{5} + 119 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 9 T - 49 T^{2} + 108 T^{3} + 8640 T^{4} + 108 p T^{5} - 49 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 18 T + 225 T^{2} + 2106 T^{3} + 16436 T^{4} + 2106 p T^{5} + 225 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 19 T + 137 T^{2} + 1558 T^{3} + 19504 T^{4} + 1558 p T^{5} + 137 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^3$ | \( 1 - 113 T^{2} + 7440 T^{4} - 113 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 6 T - 3 T^{2} + 90 T^{3} - 5068 T^{4} + 90 p T^{5} - 3 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 - 154 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 9 T - 109 T^{2} + 108 T^{3} + 20970 T^{4} + 108 p T^{5} - 109 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 27 T + 473 T^{2} - 6210 T^{3} + 67062 T^{4} - 6210 p T^{5} + 473 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.17285942522205425486296098798, −7.12881555869283679029830742699, −6.71938448756814750981145849552, −6.23202321218768358836354665819, −6.15755990880469641101682424949, −5.96815269680143070629186166598, −5.92529115743043858493509836800, −5.88011070414786089192942119330, −5.56020265099271106505085623521, −5.14559467704018874175874733711, −4.93302925441963677960241402278, −4.80223509625920338996651104558, −4.53952548738289254363154001466, −4.50325969159922812353089189089, −3.63345108687961436578393319518, −3.36061022765361375452663154638, −3.26969318866354942071206103045, −3.08600351985784616355767759746, −2.92498393573560519748166763561, −2.54393436579095325077830446798, −1.96259242993165211553045965461, −1.76467755419952867882143591227, −1.20312932012362480941499771938, −0.905688510322046202803204416781, −0.45964583906617574729050373151,
0.45964583906617574729050373151, 0.905688510322046202803204416781, 1.20312932012362480941499771938, 1.76467755419952867882143591227, 1.96259242993165211553045965461, 2.54393436579095325077830446798, 2.92498393573560519748166763561, 3.08600351985784616355767759746, 3.26969318866354942071206103045, 3.36061022765361375452663154638, 3.63345108687961436578393319518, 4.50325969159922812353089189089, 4.53952548738289254363154001466, 4.80223509625920338996651104558, 4.93302925441963677960241402278, 5.14559467704018874175874733711, 5.56020265099271106505085623521, 5.88011070414786089192942119330, 5.92529115743043858493509836800, 5.96815269680143070629186166598, 6.15755990880469641101682424949, 6.23202321218768358836354665819, 6.71938448756814750981145849552, 7.12881555869283679029830742699, 7.17285942522205425486296098798
