| L(s) = 1 | + 4·4-s + 5·7-s − 3·9-s + 12·16-s + 20·28-s − 12·36-s − 20·37-s + 10·43-s + 18·49-s − 15·63-s + 32·64-s − 10·67-s − 8·79-s + 9·81-s + 38·109-s + 60·112-s + 22·121-s + 127-s + 131-s + 137-s + 139-s − 36·144-s − 80·148-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 2·4-s + 1.88·7-s − 9-s + 3·16-s + 3.77·28-s − 2·36-s − 3.28·37-s + 1.52·43-s + 18/7·49-s − 1.88·63-s + 4·64-s − 1.22·67-s − 0.900·79-s + 81-s + 3.63·109-s + 5.66·112-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3·144-s − 6.57·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.584580724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.584580724\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−11.10842878759503743210556032085, −10.77561790762563975980625846698, −10.37275353067317937034519186629, −10.13517445056906476310256448150, −9.145408195227827669094909749635, −8.634713223217658252321103660565, −8.449740893258303521461026460018, −7.82061418851378274455513592247, −7.32759021226586969853079106367, −7.28778582772320135952983078789, −6.48808743520627539616072849317, −6.03813736966547826216118988599, −5.46228399035234945665756082909, −5.25001449412478868326130679873, −4.49607333079499478568509703407, −3.63581623058477547447572411624, −3.10279454558850298002594662363, −2.34562082576932305220409514843, −1.92888964427847317577486867770, −1.24972451968203369021735875245,
1.24972451968203369021735875245, 1.92888964427847317577486867770, 2.34562082576932305220409514843, 3.10279454558850298002594662363, 3.63581623058477547447572411624, 4.49607333079499478568509703407, 5.25001449412478868326130679873, 5.46228399035234945665756082909, 6.03813736966547826216118988599, 6.48808743520627539616072849317, 7.28778582772320135952983078789, 7.32759021226586969853079106367, 7.82061418851378274455513592247, 8.449740893258303521461026460018, 8.634713223217658252321103660565, 9.145408195227827669094909749635, 10.13517445056906476310256448150, 10.37275353067317937034519186629, 10.77561790762563975980625846698, 11.10842878759503743210556032085
