| L(s) = 1 | + 2·3-s − 4-s + 2·7-s + 3·9-s − 8·11-s − 2·12-s − 8·13-s − 3·16-s + 4·17-s + 4·21-s + 2·23-s + 4·27-s − 2·28-s − 6·29-s − 16·33-s − 3·36-s − 16·37-s − 16·39-s − 6·41-s + 16·43-s + 8·44-s + 2·47-s − 6·48-s − 11·49-s + 8·51-s + 8·52-s − 4·53-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s + 0.755·7-s + 9-s − 2.41·11-s − 0.577·12-s − 2.21·13-s − 3/4·16-s + 0.970·17-s + 0.872·21-s + 0.417·23-s + 0.769·27-s − 0.377·28-s − 1.11·29-s − 2.78·33-s − 1/2·36-s − 2.63·37-s − 2.56·39-s − 0.937·41-s + 2.43·43-s + 1.20·44-s + 0.291·47-s − 0.866·48-s − 1.57·49-s + 1.12·51-s + 1.10·52-s − 0.549·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12425625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12425625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 47 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 8 T + 3 p T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 35 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 55 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 16 T + 126 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 79 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 16 T + 138 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 20 T + 191 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2 T + 75 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 119 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 134 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 20 T + 266 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 186 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.217198380165869182747558166138, −8.123872592142635482826933870250, −7.52263222634567871047367943775, −7.43751662001673011341206139141, −7.29745520745495085110965811944, −6.78585614540459017253345756543, −6.01445117065334862304234225611, −5.54943705204085188520559461394, −5.14591045499074417122822406934, −5.01627813261020809880797425646, −4.54182584650991107472891690243, −4.36170090772263821306866599562, −3.44135174968241146884130943885, −3.22068974932638116197910080289, −2.75136717041583464148880957275, −2.27957819533470281315089801318, −2.00646790922642152845075487646, −1.35971850042607192829234392338, 0, 0,
1.35971850042607192829234392338, 2.00646790922642152845075487646, 2.27957819533470281315089801318, 2.75136717041583464148880957275, 3.22068974932638116197910080289, 3.44135174968241146884130943885, 4.36170090772263821306866599562, 4.54182584650991107472891690243, 5.01627813261020809880797425646, 5.14591045499074417122822406934, 5.54943705204085188520559461394, 6.01445117065334862304234225611, 6.78585614540459017253345756543, 7.29745520745495085110965811944, 7.43751662001673011341206139141, 7.52263222634567871047367943775, 8.123872592142635482826933870250, 8.217198380165869182747558166138
