| L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 5-s − 2·6-s + 9-s − 2·10-s − 2·12-s + 15-s − 4·16-s + 2·18-s − 2·20-s + 25-s − 27-s + 2·30-s − 20·31-s − 8·32-s + 2·36-s − 45-s + 4·48-s − 10·49-s + 2·50-s + 20·53-s − 2·54-s + 2·60-s − 40·62-s − 8·64-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s − 0.816·6-s + 1/3·9-s − 0.632·10-s − 0.577·12-s + 0.258·15-s − 16-s + 0.471·18-s − 0.447·20-s + 1/5·25-s − 0.192·27-s + 0.365·30-s − 3.59·31-s − 1.41·32-s + 1/3·36-s − 0.149·45-s + 0.577·48-s − 1.42·49-s + 0.282·50-s + 2.74·53-s − 0.272·54-s + 0.258·60-s − 5.08·62-s − 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$ | \( 1 + T \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.796347544631119103074783647720, −8.385333539337096037266712335397, −7.47453418205941725695143179171, −7.27017097594245566397215443793, −6.84957657199386693405152566680, −6.16619991592203718868276685129, −5.68018452289742730764555051785, −5.34999487270269556130717190838, −4.87764075487564301030729914879, −4.11875786665161050620939976019, −3.85363731820904269766144976313, −3.26464349646293471940721383930, −2.45982269552428865640106975011, −1.61170310790947852080742475943, 0,
1.61170310790947852080742475943, 2.45982269552428865640106975011, 3.26464349646293471940721383930, 3.85363731820904269766144976313, 4.11875786665161050620939976019, 4.87764075487564301030729914879, 5.34999487270269556130717190838, 5.68018452289742730764555051785, 6.16619991592203718868276685129, 6.84957657199386693405152566680, 7.27017097594245566397215443793, 7.47453418205941725695143179171, 8.385333539337096037266712335397, 8.796347544631119103074783647720
