| L(s) = 1 | − 4·3-s − 2·7-s + 6·9-s − 11-s − 2·13-s − 6·17-s − 8·19-s + 8·21-s + 25-s + 4·27-s + 4·31-s + 4·33-s + 4·37-s + 8·39-s + 12·41-s − 2·43-s − 2·49-s + 24·51-s − 12·53-s + 32·57-s + 4·61-s − 12·63-s − 8·67-s − 24·71-s − 14·73-s − 4·75-s + 2·77-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 0.755·7-s + 2·9-s − 0.301·11-s − 0.554·13-s − 1.45·17-s − 1.83·19-s + 1.74·21-s + 1/5·25-s + 0.769·27-s + 0.718·31-s + 0.696·33-s + 0.657·37-s + 1.28·39-s + 1.87·41-s − 0.304·43-s − 2/7·49-s + 3.36·51-s − 1.64·53-s + 4.23·57-s + 0.512·61-s − 1.51·63-s − 0.977·67-s − 2.84·71-s − 1.63·73-s − 0.461·75-s + 0.227·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.7324805054, −17.3081878781, −17.1597632045, −16.4140234568, −16.0995603957, −15.7352152522, −14.7919331095, −14.5613026708, −13.4238640951, −13.0023664158, −12.5644053833, −11.9941542857, −11.5085445320, −10.8347395030, −10.7104157084, −10.0329245520, −9.15115844234, −8.55221755078, −7.52651413140, −6.57891116466, −6.30624098912, −5.84388404673, −4.78130792718, −4.43743797714, −2.67336507985, 0,
2.67336507985, 4.43743797714, 4.78130792718, 5.84388404673, 6.30624098912, 6.57891116466, 7.52651413140, 8.55221755078, 9.15115844234, 10.0329245520, 10.7104157084, 10.8347395030, 11.5085445320, 11.9941542857, 12.5644053833, 13.0023664158, 13.4238640951, 14.5613026708, 14.7919331095, 15.7352152522, 16.0995603957, 16.4140234568, 17.1597632045, 17.3081878781, 17.7324805054
