| L(s) = 1 | + 2·3-s + 4-s + 3·9-s + 2·12-s + 16-s − 12·23-s − 10·25-s + 4·27-s + 4·31-s + 3·36-s − 8·37-s + 12·47-s + 2·48-s + 2·49-s + 12·53-s − 24·59-s + 64-s + 16·67-s − 24·69-s − 20·75-s + 5·81-s − 12·89-s − 12·92-s + 8·93-s − 20·97-s − 10·100-s − 20·103-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s + 9-s + 0.577·12-s + 1/4·16-s − 2.50·23-s − 2·25-s + 0.769·27-s + 0.718·31-s + 1/2·36-s − 1.31·37-s + 1.75·47-s + 0.288·48-s + 2/7·49-s + 1.64·53-s − 3.12·59-s + 1/8·64-s + 1.95·67-s − 2.88·69-s − 2.30·75-s + 5/9·81-s − 1.27·89-s − 1.25·92-s + 0.829·93-s − 2.03·97-s − 100-s − 1.97·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1572516 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1572516 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−7.902026628622195022008075969731, −7.36542118194181532299559628639, −6.84732233829542762166868447522, −6.48446256583549422713499573426, −5.90440030555350655192035516561, −5.61719283222003359694793802526, −5.06506150537095507131369969579, −4.17114509058355439704198819237, −3.89674503164637365373404147869, −3.80497823702111438859005699796, −2.72251484544452322835217288408, −2.57951876002705252224113393018, −1.86863222374205151449360817840, −1.43497668093600715577910248029, 0,
1.43497668093600715577910248029, 1.86863222374205151449360817840, 2.57951876002705252224113393018, 2.72251484544452322835217288408, 3.80497823702111438859005699796, 3.89674503164637365373404147869, 4.17114509058355439704198819237, 5.06506150537095507131369969579, 5.61719283222003359694793802526, 5.90440030555350655192035516561, 6.48446256583549422713499573426, 6.84732233829542762166868447522, 7.36542118194181532299559628639, 7.902026628622195022008075969731
