| L(s) = 1 | − 6·9-s − 2·11-s − 8·17-s + 25-s + 12·41-s − 4·43-s − 10·49-s − 24·59-s + 24·67-s + 8·73-s + 27·81-s − 4·83-s + 12·89-s − 4·97-s + 12·99-s + 12·107-s + 28·113-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 48·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 2·9-s − 0.603·11-s − 1.94·17-s + 1/5·25-s + 1.87·41-s − 0.609·43-s − 1.42·49-s − 3.12·59-s + 2.93·67-s + 0.936·73-s + 3·81-s − 0.439·83-s + 1.27·89-s − 0.406·97-s + 1.20·99-s + 1.16·107-s + 2.63·113-s + 3/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 3.88·153-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3097600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3097600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−7.36247134612527590927307037370, −6.85667269899173093324578324851, −6.30961469790096626723393594297, −6.21087114261569312686272580596, −5.72971066068134741309412572095, −5.17000472187588479255187728599, −4.83284777784030043470160695571, −4.45821261643777188910209572887, −3.79836956390090775269445181604, −3.12595338350487188300141328594, −2.96374376382014471097320958505, −2.17666388867997363005053217178, −2.03399759604601396643465282622, −0.71531896275518796286339281459, 0,
0.71531896275518796286339281459, 2.03399759604601396643465282622, 2.17666388867997363005053217178, 2.96374376382014471097320958505, 3.12595338350487188300141328594, 3.79836956390090775269445181604, 4.45821261643777188910209572887, 4.83284777784030043470160695571, 5.17000472187588479255187728599, 5.72971066068134741309412572095, 6.21087114261569312686272580596, 6.30961469790096626723393594297, 6.85667269899173093324578324851, 7.36247134612527590927307037370
