| L(s) = 1 | − 2·3-s − 3·9-s + 4·13-s − 4·16-s − 4·17-s − 2·23-s − 9·25-s + 14·27-s − 8·39-s − 12·43-s + 8·48-s − 10·49-s + 8·51-s − 12·53-s + 24·61-s + 4·69-s + 18·75-s − 20·79-s − 4·81-s + 4·101-s − 32·103-s + 36·107-s + 18·113-s − 12·117-s + 121-s + 127-s + 24·129-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 9-s + 1.10·13-s − 16-s − 0.970·17-s − 0.417·23-s − 9/5·25-s + 2.69·27-s − 1.28·39-s − 1.82·43-s + 1.15·48-s − 1.42·49-s + 1.12·51-s − 1.64·53-s + 3.07·61-s + 0.481·69-s + 2.07·75-s − 2.25·79-s − 4/9·81-s + 0.398·101-s − 3.15·103-s + 3.48·107-s + 1.69·113-s − 1.10·117-s + 1/11·121-s + 0.0887·127-s + 2.11·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20449 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20449 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−10.91113543903689003112366466403, −10.03550909718107888433464868208, −9.697960688174592536443811508987, −8.679083346749551349707825193313, −8.603539619290756001226038948684, −7.972094664118160382294418175959, −6.98822017919486224475963260290, −6.36261389471308870138602900888, −6.13551737647869362536665067533, −5.42467290200223196482028296291, −4.84072493234829119227722123670, −4.01156608984212096591011250141, −3.11506443817379691538930485019, −1.98576135654663457689533375342, 0,
1.98576135654663457689533375342, 3.11506443817379691538930485019, 4.01156608984212096591011250141, 4.84072493234829119227722123670, 5.42467290200223196482028296291, 6.13551737647869362536665067533, 6.36261389471308870138602900888, 6.98822017919486224475963260290, 7.972094664118160382294418175959, 8.603539619290756001226038948684, 8.679083346749551349707825193313, 9.697960688174592536443811508987, 10.03550909718107888433464868208, 10.91113543903689003112366466403
