| L(s) = 1 | − 3-s − 5-s − 2·9-s + 15-s − 4·16-s − 2·23-s − 4·25-s + 5·27-s + 14·31-s + 2·45-s − 16·47-s + 4·48-s − 10·49-s + 12·53-s + 2·69-s + 4·75-s + 4·80-s + 81-s − 14·93-s + 18·113-s + 2·115-s + 9·125-s + 127-s + 131-s − 5·135-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 2/3·9-s + 0.258·15-s − 16-s − 0.417·23-s − 4/5·25-s + 0.962·27-s + 2.51·31-s + 0.298·45-s − 2.33·47-s + 0.577·48-s − 1.42·49-s + 1.64·53-s + 0.240·69-s + 0.461·75-s + 0.447·80-s + 1/9·81-s − 1.45·93-s + 1.69·113-s + 0.186·115-s + 0.804·125-s + 0.0887·127-s + 0.0873·131-s − 0.430·135-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + T + p T^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−7.14519786679342819384976248425, −6.89527691908007437702093531710, −6.36653146481970975334128540406, −6.15517988443792192476449132837, −5.73628449711830132991006753856, −5.10459545607334527318352567687, −4.71920964949621529052226248224, −4.50304996929188052023847279147, −3.84792502687481147981679237100, −3.33733559656066460416950662826, −2.82767102164583012351673409929, −2.32159156072485424081104661544, −1.65668244707724998022653476825, −0.74979793610769007127710514804, 0,
0.74979793610769007127710514804, 1.65668244707724998022653476825, 2.32159156072485424081104661544, 2.82767102164583012351673409929, 3.33733559656066460416950662826, 3.84792502687481147981679237100, 4.50304996929188052023847279147, 4.71920964949621529052226248224, 5.10459545607334527318352567687, 5.73628449711830132991006753856, 6.15517988443792192476449132837, 6.36653146481970975334128540406, 6.89527691908007437702093531710, 7.14519786679342819384976248425
