| L(s) = 1 | − 4-s + 4·13-s + 16-s + 4·17-s + 12·23-s + 6·25-s + 8·43-s − 49-s − 4·52-s − 8·53-s + 24·61-s − 64-s − 4·68-s − 12·92-s − 6·100-s − 4·101-s + 28·103-s + 24·107-s − 28·113-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.10·13-s + 1/4·16-s + 0.970·17-s + 2.50·23-s + 6/5·25-s + 1.21·43-s − 1/7·49-s − 0.554·52-s − 1.09·53-s + 3.07·61-s − 1/8·64-s − 0.485·68-s − 1.25·92-s − 3/5·100-s − 0.398·101-s + 2.75·103-s + 2.32·107-s − 2.63·113-s + 2·121-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 + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.859334456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.859334456\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−9.424698992451957286159519272060, −9.152418986267061821224835424456, −8.749168164747724714042925181899, −8.595071690056888259044084140301, −7.934325375544616487882788847408, −7.75638654433286324095934684822, −7.06286264566425336447786157760, −6.84202525551738398982228498201, −6.45353718205156970366882323021, −5.77219638747517919045293616973, −5.52967677766559342397443899790, −5.04112035904256504823063621828, −4.66830019815387131361063796642, −4.16199777834455602780680036225, −3.50908506190115822744657955592, −3.24666959081167690420024510323, −2.76494907188270627237837438418, −1.97509705869079714685717712568, −0.977408154245149686882164570202, −0.924820628258119128540482393297,
0.924820628258119128540482393297, 0.977408154245149686882164570202, 1.97509705869079714685717712568, 2.76494907188270627237837438418, 3.24666959081167690420024510323, 3.50908506190115822744657955592, 4.16199777834455602780680036225, 4.66830019815387131361063796642, 5.04112035904256504823063621828, 5.52967677766559342397443899790, 5.77219638747517919045293616973, 6.45353718205156970366882323021, 6.84202525551738398982228498201, 7.06286264566425336447786157760, 7.75638654433286324095934684822, 7.934325375544616487882788847408, 8.595071690056888259044084140301, 8.749168164747724714042925181899, 9.152418986267061821224835424456, 9.424698992451957286159519272060
