| L(s) = 1 | + 2·2-s − 2·3-s − 4-s + 5-s − 4·6-s − 2·7-s − 8·8-s + 3·9-s + 2·10-s + 2·12-s − 2·13-s − 4·14-s − 2·15-s − 7·16-s − 9·17-s + 6·18-s − 3·19-s − 20-s + 4·21-s + 5·23-s + 16·24-s + 2·25-s − 4·26-s − 4·27-s + 2·28-s + 4·29-s − 4·30-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s − 1/2·4-s + 0.447·5-s − 1.63·6-s − 0.755·7-s − 2.82·8-s + 9-s + 0.632·10-s + 0.577·12-s − 0.554·13-s − 1.06·14-s − 0.516·15-s − 7/4·16-s − 2.18·17-s + 1.41·18-s − 0.688·19-s − 0.223·20-s + 0.872·21-s + 1.04·23-s + 3.26·24-s + 2/5·25-s − 0.784·26-s − 0.769·27-s + 0.377·28-s + 0.742·29-s − 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6456681 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6456681 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.182736475\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.182736475\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $D_{4}$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 9 T + 43 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 39 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 5 T + 21 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 13 T + 85 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 15 T + 127 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 82 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 2 T - 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 18 T + 194 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 6 T + 150 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 9 T + 167 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 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
−9.205226063654288417173313386633, −8.816316193648678614747280199707, −8.281850049176596006050092527735, −8.233263839458746192188372506956, −7.16377255829697943252829319920, −6.84856723572555738168162051153, −6.53489529752984582730625673075, −6.48970423116687076905399180688, −5.68291193655437884592204819590, −5.58624334784972556092541957129, −5.10823645808415181861443452513, −4.82121280744219104749810978658, −4.44146216307377654473688137912, −3.98796623431240742392134779001, −3.74790328066747404480917589662, −3.09941777661431741547365958866, −2.33333190707380805847933193303, −2.33303478094177573161821701729, −0.889070051038782508370334149570, −0.40658231179387011003891557820,
0.40658231179387011003891557820, 0.889070051038782508370334149570, 2.33303478094177573161821701729, 2.33333190707380805847933193303, 3.09941777661431741547365958866, 3.74790328066747404480917589662, 3.98796623431240742392134779001, 4.44146216307377654473688137912, 4.82121280744219104749810978658, 5.10823645808415181861443452513, 5.58624334784972556092541957129, 5.68291193655437884592204819590, 6.48970423116687076905399180688, 6.53489529752984582730625673075, 6.84856723572555738168162051153, 7.16377255829697943252829319920, 8.233263839458746192188372506956, 8.281850049176596006050092527735, 8.816316193648678614747280199707, 9.205226063654288417173313386633
