L(s) = 1 | + 4·3-s + 6·9-s − 10·25-s − 4·27-s + 8·31-s + 4·37-s + 24·47-s + 49-s + 12·53-s + 12·59-s + 8·67-s − 40·75-s − 37·81-s − 12·89-s + 32·93-s − 20·97-s + 8·103-s + 16·111-s + 12·113-s − 11·121-s + 127-s + 131-s + 137-s + 139-s + 96·141-s + 4·147-s + 149-s + ⋯ |
L(s) = 1 | + 2.30·3-s + 2·9-s − 2·25-s − 0.769·27-s + 1.43·31-s + 0.657·37-s + 3.50·47-s + 1/7·49-s + 1.64·53-s + 1.56·59-s + 0.977·67-s − 4.61·75-s − 4.11·81-s − 1.27·89-s + 3.31·93-s − 2.03·97-s + 0.788·103-s + 1.51·111-s + 1.12·113-s − 121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 8.08·141-s + 0.329·147-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1517824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1517824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.751067554\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.751067554\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−8.182643116839158569055135937831, −7.52725309122624587441708234050, −7.29668061464756137953129595214, −6.87975810404488340043398375902, −6.10121749499441085588030360800, −5.62168100149868099368912779374, −5.51487510751237752431573865412, −4.42789561057506381074928638435, −4.05731801132236751489021266456, −3.82418745638366191622617161116, −3.20671223694547121384446442813, −2.49822445009788286784730652330, −2.48514181668809031419660323366, −1.86250960874307840989277735252, −0.795913931859384347135515195825,
0.795913931859384347135515195825, 1.86250960874307840989277735252, 2.48514181668809031419660323366, 2.49822445009788286784730652330, 3.20671223694547121384446442813, 3.82418745638366191622617161116, 4.05731801132236751489021266456, 4.42789561057506381074928638435, 5.51487510751237752431573865412, 5.62168100149868099368912779374, 6.10121749499441085588030360800, 6.87975810404488340043398375902, 7.29668061464756137953129595214, 7.52725309122624587441708234050, 8.182643116839158569055135937831