| L(s) = 1 | + 8·7-s − 3·9-s + 4·13-s + 8·19-s + 25-s + 16·31-s − 12·37-s − 16·43-s + 34·49-s + 4·61-s − 24·63-s + 16·67-s − 12·73-s + 9·81-s + 32·91-s − 28·97-s − 8·103-s − 28·109-s − 12·117-s − 6·121-s + 127-s + 131-s + 64·133-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 3.02·7-s − 9-s + 1.10·13-s + 1.83·19-s + 1/5·25-s + 2.87·31-s − 1.97·37-s − 2.43·43-s + 34/7·49-s + 0.512·61-s − 3.02·63-s + 1.95·67-s − 1.40·73-s + 81-s + 3.35·91-s − 2.84·97-s − 0.788·103-s − 2.68·109-s − 1.10·117-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 5.54·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.460509320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.460509320\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−8.187189248591461729679142296814, −8.094484908266087825781152713303, −7.38986169737030489086018454093, −6.81982863783073968912914119025, −6.52411082007979697711723996792, −5.61529786752672156928392151688, −5.34437678528517976086556923571, −5.16731330798328126955314301729, −4.55607707806255331316875895933, −4.16841981012504174266678012358, −3.31415635628336508949565358363, −2.94471466186386712232384138589, −2.05937366765957406711596802667, −1.46465509347525672726427035614, −1.01814837175580913037691321841,
1.01814837175580913037691321841, 1.46465509347525672726427035614, 2.05937366765957406711596802667, 2.94471466186386712232384138589, 3.31415635628336508949565358363, 4.16841981012504174266678012358, 4.55607707806255331316875895933, 5.16731330798328126955314301729, 5.34437678528517976086556923571, 5.61529786752672156928392151688, 6.52411082007979697711723996792, 6.81982863783073968912914119025, 7.38986169737030489086018454093, 8.094484908266087825781152713303, 8.187189248591461729679142296814
