| L(s) = 1 | − 2·3-s + 3·4-s + 3·9-s − 6·12-s + 5·16-s − 4·17-s + 6·25-s − 4·27-s − 20·29-s + 9·36-s + 24·43-s − 10·48-s − 2·49-s + 8·51-s + 12·53-s − 4·61-s + 3·64-s − 12·68-s − 12·75-s + 16·79-s + 5·81-s + 40·87-s + 18·100-s + 36·101-s + 24·107-s − 12·108-s − 12·113-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 3/2·4-s + 9-s − 1.73·12-s + 5/4·16-s − 0.970·17-s + 6/5·25-s − 0.769·27-s − 3.71·29-s + 3/2·36-s + 3.65·43-s − 1.44·48-s − 2/7·49-s + 1.12·51-s + 1.64·53-s − 0.512·61-s + 3/8·64-s − 1.45·68-s − 1.38·75-s + 1.80·79-s + 5/9·81-s + 4.28·87-s + 9/5·100-s + 3.58·101-s + 2.32·107-s − 1.15·108-s − 1.12·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.608765409\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.608765409\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 174 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−11.15988164067414791761595922941, −10.79091640292724697445636220594, −10.66282457223276246036277091316, −9.949140370334661451655029102257, −9.328119828160634463505411049130, −9.094854179047065514979181005699, −8.475039207531789124869526032260, −7.47885806748749413135659960307, −7.39656934678084437616839024501, −7.20213804215951493017369447649, −6.45235019367160744624712390795, −5.95006734907181090512861581927, −5.83579496331200400024470915421, −5.16203985282114134813121088839, −4.49955788025706329930696655792, −3.89621577422803255855213848135, −3.25291631260208456149630665964, −2.20603626271941433523197019764, −2.03105277345741485102557666130, −0.803801514756463595774766698493,
0.803801514756463595774766698493, 2.03105277345741485102557666130, 2.20603626271941433523197019764, 3.25291631260208456149630665964, 3.89621577422803255855213848135, 4.49955788025706329930696655792, 5.16203985282114134813121088839, 5.83579496331200400024470915421, 5.95006734907181090512861581927, 6.45235019367160744624712390795, 7.20213804215951493017369447649, 7.39656934678084437616839024501, 7.47885806748749413135659960307, 8.475039207531789124869526032260, 9.094854179047065514979181005699, 9.328119828160634463505411049130, 9.949140370334661451655029102257, 10.66282457223276246036277091316, 10.79091640292724697445636220594, 11.15988164067414791761595922941
