| L(s) = 1 | − 2-s + 4-s − 8-s + 3·11-s + 13-s + 16-s − 3·22-s + 6·23-s − 25-s − 26-s − 32-s − 5·37-s + 3·44-s − 6·46-s − 15·47-s − 4·49-s + 50-s + 52-s + 15·59-s − 11·61-s + 64-s − 9·71-s + 13·73-s + 5·74-s − 6·83-s − 3·88-s + 6·92-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 0.904·11-s + 0.277·13-s + 1/4·16-s − 0.639·22-s + 1.25·23-s − 1/5·25-s − 0.196·26-s − 0.176·32-s − 0.821·37-s + 0.452·44-s − 0.884·46-s − 2.18·47-s − 4/7·49-s + 0.141·50-s + 0.138·52-s + 1.95·59-s − 1.40·61-s + 1/8·64-s − 1.06·71-s + 1.52·73-s + 0.581·74-s − 0.658·83-s − 0.319·88-s + 0.625·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7241670961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7241670961\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 124 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−11.70422507635601315913629872862, −11.09685192185071534269752296548, −10.76223527756245710310430313431, −9.881598400770791938343277323819, −9.553643947868349165298296376119, −8.867656109559051004177636489278, −8.404039025114582906179535096937, −7.75273388820878922793671315214, −6.83288923769794203043070161468, −6.65973926664735244969619365533, −5.68713959385716354446117983973, −4.89225534460440060165371312525, −3.86250849424392473154621535990, −2.97384047821442421624124554305, −1.54416781605207259588976454324,
1.54416781605207259588976454324, 2.97384047821442421624124554305, 3.86250849424392473154621535990, 4.89225534460440060165371312525, 5.68713959385716354446117983973, 6.65973926664735244969619365533, 6.83288923769794203043070161468, 7.75273388820878922793671315214, 8.404039025114582906179535096937, 8.867656109559051004177636489278, 9.553643947868349165298296376119, 9.881598400770791938343277323819, 10.76223527756245710310430313431, 11.09685192185071534269752296548, 11.70422507635601315913629872862
