| L(s) = 1 | − 4-s − 2·7-s + 10·13-s − 3·16-s − 2·19-s + 2·25-s + 2·28-s + 10·31-s − 2·37-s − 2·43-s − 11·49-s − 10·52-s + 4·61-s + 7·64-s + 16·67-s + 4·73-s + 2·76-s − 2·79-s − 20·91-s + 34·97-s − 2·100-s + 16·103-s + 34·109-s + 6·112-s − 10·121-s − 10·124-s + 127-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.755·7-s + 2.77·13-s − 3/4·16-s − 0.458·19-s + 2/5·25-s + 0.377·28-s + 1.79·31-s − 0.328·37-s − 0.304·43-s − 1.57·49-s − 1.38·52-s + 0.512·61-s + 7/8·64-s + 1.95·67-s + 0.468·73-s + 0.229·76-s − 0.225·79-s − 2.09·91-s + 3.45·97-s − 1/5·100-s + 1.57·103-s + 3.25·109-s + 0.566·112-s − 0.909·121-s − 0.898·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.213964138\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.213964138\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 118 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.60055799547426009652810274468, −11.68613044632300397194764208501, −11.27363637433942366154952747032, −11.14495484165665719309580933705, −10.21723916378989491163277649923, −10.13654896803195556492580659962, −9.388168687038182801682601425010, −8.833245131673608769163171550514, −8.519652151235185896572998162213, −8.223590322904053967862384369796, −7.37225555795780709109469246797, −6.51667717431085117725053286253, −6.38988047348155697162740997975, −5.94367436151325889775645239570, −4.95763268889965373286462212722, −4.50551446413152782472606775083, −3.52168957571220695631214088920, −3.47172281941875945571553746616, −2.21095194874431803639976263612, −0.970090806857385628640518965887,
0.970090806857385628640518965887, 2.21095194874431803639976263612, 3.47172281941875945571553746616, 3.52168957571220695631214088920, 4.50551446413152782472606775083, 4.95763268889965373286462212722, 5.94367436151325889775645239570, 6.38988047348155697162740997975, 6.51667717431085117725053286253, 7.37225555795780709109469246797, 8.223590322904053967862384369796, 8.519652151235185896572998162213, 8.833245131673608769163171550514, 9.388168687038182801682601425010, 10.13654896803195556492580659962, 10.21723916378989491163277649923, 11.14495484165665719309580933705, 11.27363637433942366154952747032, 11.68613044632300397194764208501, 12.60055799547426009652810274468
