L(s) = 1 | − 8·7-s + 4·23-s − 25-s − 16·31-s + 4·41-s + 20·47-s + 34·49-s − 24·71-s − 28·73-s − 16·79-s − 36·89-s − 12·97-s − 32·103-s + 40·113-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 32·161-s + 163-s + 167-s + 22·169-s + 173-s + ⋯ |
L(s) = 1 | − 3.02·7-s + 0.834·23-s − 1/5·25-s − 2.87·31-s + 0.624·41-s + 2.91·47-s + 34/7·49-s − 2.84·71-s − 3.27·73-s − 1.80·79-s − 3.81·89-s − 1.21·97-s − 3.15·103-s + 3.76·113-s + 6/11·121-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 + 0.0798·157-s − 2.52·161-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
good | 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−8.694075923882302512842694211987, −8.575959756909580328537814441038, −7.37442523734176563416851148320, −7.35281773056447823465597169468, −7.16896869400984052141077865702, −6.86849658158331012564599294703, −6.14327578343807227443782365367, −5.88721366877117713622398204739, −5.69786403664708467998158248543, −5.41454715346349893808092207943, −4.29962952503001469251322412214, −4.29157023170158336192993164055, −3.76477967976697913435000135753, −3.22174139329960197830469308146, −2.85782607181659317622961978511, −2.73327454524921032577164931679, −1.82615454885304414205704042256, −1.11359543999282860657323264902, 0, 0,
1.11359543999282860657323264902, 1.82615454885304414205704042256, 2.73327454524921032577164931679, 2.85782607181659317622961978511, 3.22174139329960197830469308146, 3.76477967976697913435000135753, 4.29157023170158336192993164055, 4.29962952503001469251322412214, 5.41454715346349893808092207943, 5.69786403664708467998158248543, 5.88721366877117713622398204739, 6.14327578343807227443782365367, 6.86849658158331012564599294703, 7.16896869400984052141077865702, 7.35281773056447823465597169468, 7.37442523734176563416851148320, 8.575959756909580328537814441038, 8.694075923882302512842694211987