| L(s) = 1 | − 3·5-s + 7-s + 9·11-s − 6·13-s − 12·17-s + 9·23-s + 5·25-s − 18·29-s − 9·31-s − 3·35-s + 2·37-s + 3·41-s − 10·43-s + 6·47-s − 6·49-s − 27·55-s − 6·59-s − 24·61-s + 18·65-s − 2·67-s + 9·77-s − 14·79-s − 6·83-s + 36·85-s + 18·89-s − 6·91-s − 12·97-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.377·7-s + 2.71·11-s − 1.66·13-s − 2.91·17-s + 1.87·23-s + 25-s − 3.34·29-s − 1.61·31-s − 0.507·35-s + 0.328·37-s + 0.468·41-s − 1.52·43-s + 0.875·47-s − 6/7·49-s − 3.64·55-s − 0.781·59-s − 3.07·61-s + 2.23·65-s − 0.244·67-s + 1.02·77-s − 1.57·79-s − 0.658·83-s + 3.90·85-s + 1.90·89-s − 0.628·91-s − 1.21·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.07944426350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07944426350\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 9 T + 38 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 18 T + 137 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 24 T + 253 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 115 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 134 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 6 T - 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 12 T + 145 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−9.161174751678670913217706957398, −8.825429849980032998087100299946, −8.818451898604271724250121376354, −7.905195555480777002153142779122, −7.54348912736802206381281372481, −7.28112402612049339777621314901, −7.01060814578107434101596984511, −6.59972199798844634021061682922, −6.32241608189410673258078795057, −5.64153391694718872080805739347, −5.10270497863213603869969747602, −4.61627555318613527602423536084, −4.42165660149710873602985285924, −3.91210225258567871081042198368, −3.70959219693282119338613136165, −3.06703515336781640809235490266, −2.43863365954322535998482304840, −1.65608631205647557733701236903, −1.52848527643472662795502894987, −0.095014008933280367276803188611,
0.095014008933280367276803188611, 1.52848527643472662795502894987, 1.65608631205647557733701236903, 2.43863365954322535998482304840, 3.06703515336781640809235490266, 3.70959219693282119338613136165, 3.91210225258567871081042198368, 4.42165660149710873602985285924, 4.61627555318613527602423536084, 5.10270497863213603869969747602, 5.64153391694718872080805739347, 6.32241608189410673258078795057, 6.59972199798844634021061682922, 7.01060814578107434101596984511, 7.28112402612049339777621314901, 7.54348912736802206381281372481, 7.905195555480777002153142779122, 8.818451898604271724250121376354, 8.825429849980032998087100299946, 9.161174751678670913217706957398
