| L(s) = 1 | − 6·5-s − 6·9-s + 8·13-s − 6·17-s + 17·25-s − 16·37-s + 36·45-s + 11·49-s + 12·53-s + 10·61-s − 48·65-s − 22·73-s + 27·81-s + 36·85-s − 20·89-s − 24·97-s + 4·101-s − 12·109-s − 8·113-s − 48·117-s + 3·121-s − 18·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | − 2.68·5-s − 2·9-s + 2.21·13-s − 1.45·17-s + 17/5·25-s − 2.63·37-s + 5.36·45-s + 11/7·49-s + 1.64·53-s + 1.28·61-s − 5.95·65-s − 2.57·73-s + 3·81-s + 3.90·85-s − 2.11·89-s − 2.43·97-s + 0.398·101-s − 1.14·109-s − 0.752·113-s − 4.43·117-s + 3/11·121-s − 1.60·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
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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
−7.86378097245535978247876093758, −7.18059484484420096366454394254, −6.98286792123187236566096654142, −6.56370906167522668644909842214, −5.76283520417985127913399041730, −5.67103421999051188975454804232, −5.05210681704737714861605588166, −4.14385266867940367577820086585, −4.09350585892641493698723914863, −3.71418757784612877583046899350, −3.10630100563499367104455654686, −2.78320977751625677782435169813, −1.77861406991832868392707836598, −0.65261125406809044714433414915, 0,
0.65261125406809044714433414915, 1.77861406991832868392707836598, 2.78320977751625677782435169813, 3.10630100563499367104455654686, 3.71418757784612877583046899350, 4.09350585892641493698723914863, 4.14385266867940367577820086585, 5.05210681704737714861605588166, 5.67103421999051188975454804232, 5.76283520417985127913399041730, 6.56370906167522668644909842214, 6.98286792123187236566096654142, 7.18059484484420096366454394254, 7.86378097245535978247876093758
