| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 2·5-s + 4·6-s − 2·7-s − 4·8-s + 2·9-s − 4·10-s − 6·12-s + 2·13-s + 4·14-s − 4·15-s + 5·16-s + 4·17-s − 4·18-s + 10·19-s + 6·20-s + 4·21-s + 8·23-s + 8·24-s − 2·25-s − 4·26-s − 6·27-s − 6·28-s + 8·30-s + 4·31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s − 0.755·7-s − 1.41·8-s + 2/3·9-s − 1.26·10-s − 1.73·12-s + 0.554·13-s + 1.06·14-s − 1.03·15-s + 5/4·16-s + 0.970·17-s − 0.942·18-s + 2.29·19-s + 1.34·20-s + 0.872·21-s + 1.66·23-s + 1.63·24-s − 2/5·25-s − 0.784·26-s − 1.15·27-s − 1.13·28-s + 1.46·30-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2869636 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2869636 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.125093927\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.125093927\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 138 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 126 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T - 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 16 T + 238 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−9.792013932739297530503305261404, −9.236948270889562286572797832238, −8.792009320837796494586501724492, −8.638968684562470487827204102642, −7.81389146772521843329099754063, −7.43598587136562748895795281526, −7.32124192734718837108487987529, −6.80953882986165680750760524704, −6.23991550288968214357985461373, −6.08617284403930698320911992587, −5.46321711160013896526948644038, −5.43514556741080520329185346816, −4.90603991533263024792621945557, −3.99734784755963686173452059355, −3.25589587629907182802434009235, −3.22683479759969488900065155676, −2.35044955227726418511966533150, −1.73694911276647413577540225463, −0.844655492266702785242409620114, −0.831876165024312115707329032005,
0.831876165024312115707329032005, 0.844655492266702785242409620114, 1.73694911276647413577540225463, 2.35044955227726418511966533150, 3.22683479759969488900065155676, 3.25589587629907182802434009235, 3.99734784755963686173452059355, 4.90603991533263024792621945557, 5.43514556741080520329185346816, 5.46321711160013896526948644038, 6.08617284403930698320911992587, 6.23991550288968214357985461373, 6.80953882986165680750760524704, 7.32124192734718837108487987529, 7.43598587136562748895795281526, 7.81389146772521843329099754063, 8.638968684562470487827204102642, 8.792009320837796494586501724492, 9.236948270889562286572797832238, 9.792013932739297530503305261404
