L(s) = 1 | − 8·7-s − 3·9-s − 2·13-s − 4·16-s − 6·19-s − 25-s − 16·31-s + 4·37-s − 22·43-s + 34·49-s + 28·61-s + 24·63-s − 18·67-s + 20·73-s + 20·79-s + 9·81-s + 16·91-s − 28·97-s − 12·103-s − 28·109-s + 32·112-s + 6·117-s − 6·121-s + 127-s + 131-s + 48·133-s + 137-s + ⋯ |
L(s) = 1 | − 3.02·7-s − 9-s − 0.554·13-s − 16-s − 1.37·19-s − 1/5·25-s − 2.87·31-s + 0.657·37-s − 3.35·43-s + 34/7·49-s + 3.58·61-s + 3.02·63-s − 2.19·67-s + 2.34·73-s + 2.25·79-s + 81-s + 1.67·91-s − 2.84·97-s − 1.18·103-s − 2.68·109-s + 3.02·112-s + 0.554·117-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 4.16·133-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288369 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288369 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 + p T^{2} \) |
| 179 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 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
−8.410709202677624902364232584133, −8.096748629455889385157419531865, −7.16979390191822950971725375612, −6.69315229163834238910355551443, −6.68403281767003921511335972766, −6.18510863780788689281989494770, −5.39698209592502247550241628253, −5.26927125289354544444142895540, −4.14390277306985977394407953592, −3.65455224933898053965548391576, −3.30834137302291977565583010941, −2.57742433461389850447314956434, −2.11337535992451098529704572807, 0, 0,
2.11337535992451098529704572807, 2.57742433461389850447314956434, 3.30834137302291977565583010941, 3.65455224933898053965548391576, 4.14390277306985977394407953592, 5.26927125289354544444142895540, 5.39698209592502247550241628253, 6.18510863780788689281989494770, 6.68403281767003921511335972766, 6.69315229163834238910355551443, 7.16979390191822950971725375612, 8.096748629455889385157419531865, 8.410709202677624902364232584133