L(s) = 1 | + 2·3-s + 7-s + 3·9-s − 7·11-s − 2·13-s + 17-s − 2·19-s + 2·21-s + 7·23-s + 4·27-s − 4·29-s − 6·31-s − 14·33-s + 37-s − 4·39-s − 7·41-s + 4·43-s − 6·47-s − 9·49-s + 2·51-s − 3·53-s − 4·57-s − 16·59-s − 9·61-s + 3·63-s + 12·67-s + 14·69-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.377·7-s + 9-s − 2.11·11-s − 0.554·13-s + 0.242·17-s − 0.458·19-s + 0.436·21-s + 1.45·23-s + 0.769·27-s − 0.742·29-s − 1.07·31-s − 2.43·33-s + 0.164·37-s − 0.640·39-s − 1.09·41-s + 0.609·43-s − 0.875·47-s − 9/7·49-s + 0.280·51-s − 0.412·53-s − 0.529·57-s − 2.08·59-s − 1.15·61-s + 0.377·63-s + 1.46·67-s + 1.68·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 7 | $D_{4}$ | \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 7 T + 30 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 7 T + 54 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 54 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - T - 32 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 7 T + 56 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 86 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 104 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 9 T + 104 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 3 T + 106 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 142 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 3 T + 54 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 17 T + 212 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 15 T + 144 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
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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
−7.68272922787228454529493909979, −7.58641825541479827540430036621, −7.09872079578051009473735180149, −6.84900451905353027028379191787, −6.26615140737730023667786304311, −6.03673040190530908472335232884, −5.29781195617123054747216811270, −5.23149284877696839981335537443, −4.89380252116516263380057536750, −4.58969256206134114059974237496, −4.08567840253363864292127311848, −3.55133010117424190709096055351, −3.19076589916989981262717284582, −2.95162915122110165806360903424, −2.43213820424258063885599855822, −2.23533871389667684011556812253, −1.51581194973951569214074106347, −1.31763615702739426816392650246, 0, 0,
1.31763615702739426816392650246, 1.51581194973951569214074106347, 2.23533871389667684011556812253, 2.43213820424258063885599855822, 2.95162915122110165806360903424, 3.19076589916989981262717284582, 3.55133010117424190709096055351, 4.08567840253363864292127311848, 4.58969256206134114059974237496, 4.89380252116516263380057536750, 5.23149284877696839981335537443, 5.29781195617123054747216811270, 6.03673040190530908472335232884, 6.26615140737730023667786304311, 6.84900451905353027028379191787, 7.09872079578051009473735180149, 7.58641825541479827540430036621, 7.68272922787228454529493909979