| L(s) = 1 | − 4·5-s − 9-s − 4·11-s + 11·25-s + 16·31-s + 4·41-s + 4·45-s + 10·49-s + 16·55-s + 20·59-s + 4·61-s − 24·71-s + 81-s + 20·89-s + 4·99-s − 16·101-s − 20·109-s − 10·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 64·155-s + 157-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 1/3·9-s − 1.20·11-s + 11/5·25-s + 2.87·31-s + 0.624·41-s + 0.596·45-s + 10/7·49-s + 2.15·55-s + 2.60·59-s + 0.512·61-s − 2.84·71-s + 1/9·81-s + 2.11·89-s + 0.402·99-s − 1.59·101-s − 1.91·109-s − 0.909·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 5.14·155-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7825695718\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7825695718\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−11.97373279112430878810242545265, −11.97204191946259444215266705660, −11.65746295943836746628078584697, −10.85038935669694538881413320876, −10.60982430904786714834563426049, −10.13056002929891195176822211936, −9.499311722217538316972029861289, −8.637805473386390159780841439932, −8.456148144676839489131086932562, −7.957753317329978975262304578407, −7.53127668903399757597493950167, −7.00717100446239630133784520713, −6.41014651359295177422336904223, −5.61924101980997412333828064393, −5.02781841842898122069153631094, −4.36151973355833603134652685788, −3.94695391081871525948763714951, −2.98301548675506377793302167956, −2.58342719920451962723501294913, −0.72864190327182967399520039641,
0.72864190327182967399520039641, 2.58342719920451962723501294913, 2.98301548675506377793302167956, 3.94695391081871525948763714951, 4.36151973355833603134652685788, 5.02781841842898122069153631094, 5.61924101980997412333828064393, 6.41014651359295177422336904223, 7.00717100446239630133784520713, 7.53127668903399757597493950167, 7.957753317329978975262304578407, 8.456148144676839489131086932562, 8.637805473386390159780841439932, 9.499311722217538316972029861289, 10.13056002929891195176822211936, 10.60982430904786714834563426049, 10.85038935669694538881413320876, 11.65746295943836746628078584697, 11.97204191946259444215266705660, 11.97373279112430878810242545265
