| L(s) = 1 | − 2·3-s + 2·5-s − 9-s + 2·11-s − 4·15-s − 6·17-s − 10·19-s + 2·23-s + 25-s + 6·27-s − 14·31-s − 4·33-s + 6·37-s − 8·41-s − 8·43-s − 2·45-s − 18·47-s + 12·51-s − 2·53-s + 4·55-s + 20·57-s − 2·59-s + 6·61-s + 14·67-s − 4·69-s − 16·71-s − 18·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s − 1/3·9-s + 0.603·11-s − 1.03·15-s − 1.45·17-s − 2.29·19-s + 0.417·23-s + 1/5·25-s + 1.15·27-s − 2.51·31-s − 0.696·33-s + 0.986·37-s − 1.24·41-s − 1.21·43-s − 0.298·45-s − 2.62·47-s + 1.68·51-s − 0.274·53-s + 0.539·55-s + 2.64·57-s − 0.260·59-s + 0.768·61-s + 1.71·67-s − 0.481·69-s − 1.89·71-s − 2.10·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 21 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 10 T + 61 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 29 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 14 T + 109 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 51 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 18 T + 173 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 2 T + 21 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 99 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 14 T + 165 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 18 T + 195 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 109 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 202 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−9.282603452338505939313864527875, −8.746433655073789600851677345872, −8.459778173178423228684893217369, −8.368886219650993379752570357248, −7.52669173145320873558902710923, −6.88700960964134347665508962816, −6.58279540871323665124197561269, −6.56358906699472963239017011325, −5.88031664393134143734995481761, −5.67227621051495657846197112043, −5.19479065199572185561025131632, −4.71696832469998735437972273369, −4.29590724471617649214755737223, −3.80696905151252141584302199871, −3.09420346791227010692847904113, −2.53407064111149139124857011234, −1.70895606539922509518910291691, −1.69310267289588159206378134256, 0, 0,
1.69310267289588159206378134256, 1.70895606539922509518910291691, 2.53407064111149139124857011234, 3.09420346791227010692847904113, 3.80696905151252141584302199871, 4.29590724471617649214755737223, 4.71696832469998735437972273369, 5.19479065199572185561025131632, 5.67227621051495657846197112043, 5.88031664393134143734995481761, 6.56358906699472963239017011325, 6.58279540871323665124197561269, 6.88700960964134347665508962816, 7.52669173145320873558902710923, 8.368886219650993379752570357248, 8.459778173178423228684893217369, 8.746433655073789600851677345872, 9.282603452338505939313864527875
