L(s) = 1 | − 3-s + 9-s + 8·11-s − 4·13-s + 4·17-s − 8·19-s − 6·25-s − 27-s + 12·29-s − 8·33-s + 4·39-s − 12·41-s − 7·49-s − 4·51-s − 4·53-s + 8·57-s − 4·61-s + 6·75-s − 16·79-s + 81-s − 12·87-s − 12·89-s + 8·99-s − 24·107-s − 4·117-s + 26·121-s + 12·123-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s + 2.41·11-s − 1.10·13-s + 0.970·17-s − 1.83·19-s − 6/5·25-s − 0.192·27-s + 2.22·29-s − 1.39·33-s + 0.640·39-s − 1.87·41-s − 49-s − 0.560·51-s − 0.549·53-s + 1.05·57-s − 0.512·61-s + 0.692·75-s − 1.80·79-s + 1/9·81-s − 1.28·87-s − 1.27·89-s + 0.804·99-s − 2.32·107-s − 0.369·117-s + 2.36·121-s + 1.08·123-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 677376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 677376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 + T \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.098990694093691505068092710868, −7.73989331485765301630190119608, −6.93345661243832549395930746465, −6.71233200575528481303960613231, −6.42897107072744719896577758454, −5.98762236229661263036052835320, −5.34285485329454891844739152593, −4.81379950886871885266950292817, −4.25303028692796488061253338187, −4.05302308329842801773120835994, −3.31903774124215754032548597262, −2.64453864884235939943113643892, −1.72265159921888267196112755116, −1.30073168605128255231230247487, 0,
1.30073168605128255231230247487, 1.72265159921888267196112755116, 2.64453864884235939943113643892, 3.31903774124215754032548597262, 4.05302308329842801773120835994, 4.25303028692796488061253338187, 4.81379950886871885266950292817, 5.34285485329454891844739152593, 5.98762236229661263036052835320, 6.42897107072744719896577758454, 6.71233200575528481303960613231, 6.93345661243832549395930746465, 7.73989331485765301630190119608, 8.098990694093691505068092710868