| L(s) = 1 | − 2·2-s + 2·3-s − 4-s − 2·5-s − 4·6-s − 2·7-s + 8·8-s + 3·9-s + 4·10-s − 4·11-s − 2·12-s − 2·13-s + 4·14-s − 4·15-s − 7·16-s + 2·17-s − 6·18-s + 4·19-s + 2·20-s − 4·21-s + 8·22-s + 4·23-s + 16·24-s + 4·26-s + 4·27-s + 2·28-s − 2·29-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s − 1/2·4-s − 0.894·5-s − 1.63·6-s − 0.755·7-s + 2.82·8-s + 9-s + 1.26·10-s − 1.20·11-s − 0.577·12-s − 0.554·13-s + 1.06·14-s − 1.03·15-s − 7/4·16-s + 0.485·17-s − 1.41·18-s + 0.917·19-s + 0.447·20-s − 0.872·21-s + 1.70·22-s + 0.834·23-s + 3.26·24-s + 0.784·26-s + 0.769·27-s + 0.377·28-s − 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21538881 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21538881 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 52 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 14 T + 116 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 76 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 40 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 64 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 14 T + 108 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 14 T + 176 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 134 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 160 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 150 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 182 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.133742788333491942179005659139, −7.85458584412000020384813122405, −7.55441134809861642198509331322, −7.33704961114167788734041276530, −7.07364779489091171464943033059, −6.78141647793157511741577301269, −5.64089398701553086918403205961, −5.59105105729297336088237432300, −5.06550661214212425278499767411, −4.89116370135563507429124317842, −4.14555376843798986069186701653, −3.84113492754668572312292732596, −3.62559495131015552544416817946, −3.22328732232229978114800857923, −2.45809336743100677290017638711, −2.27406704740523242831823399458, −1.37223272750539278851041982178, −1.00992679026079716066102961188, 0, 0,
1.00992679026079716066102961188, 1.37223272750539278851041982178, 2.27406704740523242831823399458, 2.45809336743100677290017638711, 3.22328732232229978114800857923, 3.62559495131015552544416817946, 3.84113492754668572312292732596, 4.14555376843798986069186701653, 4.89116370135563507429124317842, 5.06550661214212425278499767411, 5.59105105729297336088237432300, 5.64089398701553086918403205961, 6.78141647793157511741577301269, 7.07364779489091171464943033059, 7.33704961114167788734041276530, 7.55441134809861642198509331322, 7.85458584412000020384813122405, 8.133742788333491942179005659139
