| L(s) = 1 | − 3-s − 5-s + 3·9-s − 3·11-s + 12·13-s + 15-s − 5·17-s + 19-s + 7·23-s + 5·25-s − 8·27-s + 4·29-s − 5·31-s + 3·33-s − 3·37-s − 12·39-s + 4·41-s − 8·43-s − 3·45-s + 5·47-s + 5·51-s + 53-s + 3·55-s − 57-s + 15·59-s − 5·61-s − 12·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 9-s − 0.904·11-s + 3.32·13-s + 0.258·15-s − 1.21·17-s + 0.229·19-s + 1.45·23-s + 25-s − 1.53·27-s + 0.742·29-s − 0.898·31-s + 0.522·33-s − 0.493·37-s − 1.92·39-s + 0.624·41-s − 1.21·43-s − 0.447·45-s + 0.729·47-s + 0.700·51-s + 0.137·53-s + 0.404·55-s − 0.132·57-s + 1.95·59-s − 0.640·61-s − 1.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.424240895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.424240895\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 5 T + 8 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 5 T - 22 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - T - 52 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 15 T + 166 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 9 T + 14 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 7 T - 40 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−11.21484936234559258146857679980, −11.15145070133530849482853973007, −10.73907442309271620243213136897, −10.48379510859631582914119949999, −9.752489567418408370724052408402, −9.140068152129027869207008089391, −8.694661047628741924389771571233, −8.425911655133464017826388398787, −7.901803104182270669397853063498, −7.16193884214576760508688089417, −6.71241178640570830388655050998, −6.51651468029959238085095983353, −5.53509259177546425069227481568, −5.49919240724276569573065721133, −4.59243820186855917607555370604, −4.02234528839430831765227611664, −3.60519013220142018833328961052, −2.85402239447732805081880571762, −1.69223585932844597424257716771, −0.908537998703006635301490806447,
0.908537998703006635301490806447, 1.69223585932844597424257716771, 2.85402239447732805081880571762, 3.60519013220142018833328961052, 4.02234528839430831765227611664, 4.59243820186855917607555370604, 5.49919240724276569573065721133, 5.53509259177546425069227481568, 6.51651468029959238085095983353, 6.71241178640570830388655050998, 7.16193884214576760508688089417, 7.901803104182270669397853063498, 8.425911655133464017826388398787, 8.694661047628741924389771571233, 9.140068152129027869207008089391, 9.752489567418408370724052408402, 10.48379510859631582914119949999, 10.73907442309271620243213136897, 11.15145070133530849482853973007, 11.21484936234559258146857679980
