| L(s) = 1 | + 2·3-s − 4-s + 3·9-s − 2·12-s − 6·13-s + 16-s − 4·17-s + 8·23-s + 6·25-s + 4·27-s − 20·29-s − 3·36-s − 12·39-s + 8·43-s + 2·48-s + 10·49-s − 8·51-s + 6·52-s − 12·53-s + 4·61-s − 64-s + 4·68-s + 16·69-s + 12·75-s + 5·81-s − 40·87-s − 8·92-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s + 9-s − 0.577·12-s − 1.66·13-s + 1/4·16-s − 0.970·17-s + 1.66·23-s + 6/5·25-s + 0.769·27-s − 3.71·29-s − 1/2·36-s − 1.92·39-s + 1.21·43-s + 0.288·48-s + 10/7·49-s − 1.12·51-s + 0.832·52-s − 1.64·53-s + 0.512·61-s − 1/8·64-s + 0.485·68-s + 1.92·69-s + 1.38·75-s + 5/9·81-s − 4.28·87-s − 0.834·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.069277895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.069277895\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−14.81502284481490964825941282866, −14.21173017908092884179812633889, −13.74217914458887684779145517637, −12.91154339683631396817715304873, −12.89700338987594582710171365238, −12.35987154513940471821027925503, −11.12830875965079958923609098612, −11.10594759747666845413619050874, −10.08531224531131389879503994586, −9.511202414981599377293774012324, −8.984026452504948289969072761145, −8.851745622938911644148999350723, −7.66415074492954259569590107456, −7.41831476835950984450226175229, −6.79786971902108167486082243354, −5.54761891959354108169770988651, −4.85249707345187774716120514420, −4.09590559657322186514860263713, −3.11282312486137849791066275163, −2.16401043466288981441025505835,
2.16401043466288981441025505835, 3.11282312486137849791066275163, 4.09590559657322186514860263713, 4.85249707345187774716120514420, 5.54761891959354108169770988651, 6.79786971902108167486082243354, 7.41831476835950984450226175229, 7.66415074492954259569590107456, 8.851745622938911644148999350723, 8.984026452504948289969072761145, 9.511202414981599377293774012324, 10.08531224531131389879503994586, 11.10594759747666845413619050874, 11.12830875965079958923609098612, 12.35987154513940471821027925503, 12.89700338987594582710171365238, 12.91154339683631396817715304873, 13.74217914458887684779145517637, 14.21173017908092884179812633889, 14.81502284481490964825941282866
