| L(s) = 1 | − 2·2-s − 3·3-s + 3·4-s + 5-s + 6·6-s − 4·8-s + 4·9-s − 2·10-s − 11-s − 9·12-s + 13-s − 3·15-s + 5·16-s + 12·17-s − 8·18-s − 4·19-s + 3·20-s + 2·22-s − 3·23-s + 12·24-s − 6·25-s − 2·26-s − 6·27-s + 3·29-s + 6·30-s − 3·31-s − 6·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 3/2·4-s + 0.447·5-s + 2.44·6-s − 1.41·8-s + 4/3·9-s − 0.632·10-s − 0.301·11-s − 2.59·12-s + 0.277·13-s − 0.774·15-s + 5/4·16-s + 2.91·17-s − 1.88·18-s − 0.917·19-s + 0.670·20-s + 0.426·22-s − 0.625·23-s + 2.44·24-s − 6/5·25-s − 0.392·26-s − 1.15·27-s + 0.557·29-s + 1.09·30-s − 0.538·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13147876 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13147876 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 37 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_4$ | \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T + 7 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + 19 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - T + 23 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 19 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 - 3 T + 31 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3 T + 61 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 9 T + 73 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 82 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 82 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 14 T + 154 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 3 T + 43 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 11 T + 83 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 21 T + 253 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 7 T + 11 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 20 T + 214 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 130 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.250702105335934015549438462891, −7.948020444540585025954223581384, −7.76076160882604891676030583576, −7.37706904108310094245371423553, −6.76054913774309270867412444465, −6.47328701984902667661026876569, −6.04490135670116543365577609958, −6.03000361971879034146966875015, −5.36308393485231880873654947257, −5.24439581825138048852730723536, −4.87993935682794763504726008572, −4.04248073808316820144615592023, −3.44414620999685219511328705165, −3.41380872099491922076615284091, −2.45722966882936373363677010627, −2.02961284314446325511385898272, −1.29516347236774972406017734113, −1.18900618583111964745346159764, 0, 0,
1.18900618583111964745346159764, 1.29516347236774972406017734113, 2.02961284314446325511385898272, 2.45722966882936373363677010627, 3.41380872099491922076615284091, 3.44414620999685219511328705165, 4.04248073808316820144615592023, 4.87993935682794763504726008572, 5.24439581825138048852730723536, 5.36308393485231880873654947257, 6.03000361971879034146966875015, 6.04490135670116543365577609958, 6.47328701984902667661026876569, 6.76054913774309270867412444465, 7.37706904108310094245371423553, 7.76076160882604891676030583576, 7.948020444540585025954223581384, 8.250702105335934015549438462891
