| L(s) = 1 | − 2·3-s + 3·9-s + 11-s − 12·23-s − 10·25-s − 4·27-s − 16·31-s − 2·33-s − 20·37-s + 12·47-s − 10·49-s + 8·67-s + 24·69-s − 12·71-s + 20·75-s + 5·81-s − 12·89-s + 32·93-s + 28·97-s + 3·99-s + 8·103-s + 40·111-s + 36·113-s + 121-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s + 0.301·11-s − 2.50·23-s − 2·25-s − 0.769·27-s − 2.87·31-s − 0.348·33-s − 3.28·37-s + 1.75·47-s − 1.42·49-s + 0.977·67-s + 2.88·69-s − 1.42·71-s + 2.30·75-s + 5/9·81-s − 1.27·89-s + 3.31·93-s + 2.84·97-s + 0.301·99-s + 0.788·103-s + 3.79·111-s + 3.38·113-s + 1/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3066624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3066624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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
−7.15001151228072728345472327672, −6.73373984274544568686272664379, −6.01435881870512402755326279246, −5.80771575321423739175967870457, −5.77859618909253610703452808610, −4.97589384532422545853514681972, −4.75034201814524111107553764909, −3.94115256297343253062160005188, −3.62485072881439717663109857391, −3.56342125758668518106535855673, −2.12574870769826326935834056810, −2.02292253942257951840802942475, −1.42670529206492309486707433636, 0, 0,
1.42670529206492309486707433636, 2.02292253942257951840802942475, 2.12574870769826326935834056810, 3.56342125758668518106535855673, 3.62485072881439717663109857391, 3.94115256297343253062160005188, 4.75034201814524111107553764909, 4.97589384532422545853514681972, 5.77859618909253610703452808610, 5.80771575321423739175967870457, 6.01435881870512402755326279246, 6.73373984274544568686272664379, 7.15001151228072728345472327672
