| L(s) = 1 | + 2·3-s − 4-s + 6·5-s − 3·9-s + 6·11-s − 2·12-s + 12·15-s − 3·16-s − 6·17-s − 2·19-s − 6·20-s + 17·25-s − 14·27-s + 12·33-s + 3·36-s + 4·37-s − 2·43-s − 6·44-s − 18·45-s − 6·48-s + 14·49-s − 12·51-s + 12·53-s + 36·55-s − 4·57-s − 12·60-s + 10·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s + 2.68·5-s − 9-s + 1.80·11-s − 0.577·12-s + 3.09·15-s − 3/4·16-s − 1.45·17-s − 0.458·19-s − 1.34·20-s + 17/5·25-s − 2.69·27-s + 2.08·33-s + 1/2·36-s + 0.657·37-s − 0.304·43-s − 0.904·44-s − 2.68·45-s − 0.866·48-s + 2·49-s − 1.68·51-s + 1.64·53-s + 4.85·55-s − 0.529·57-s − 1.54·60-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52441 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.549581091\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.549581091\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 229 | $C_2$ | \( 1 - 14 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 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
−13.14580004509351118819641553580, −11.75485079284358130848965550474, −11.61430493196621781706902305185, −10.99769158155824883235202256723, −10.20704899738732125371168016271, −9.933228883600645345117312716392, −9.288995754727251390007447684528, −8.998106452521941953152706182178, −8.768961470526460026351817099329, −8.587851934459013834761684572790, −7.41739122861685143280509277723, −6.71778066430661336888582716183, −6.29205951881109080234144756635, −5.78640405170751959347105161981, −5.42927634164617286025699613855, −4.38492488245376512643072179509, −3.88261708380527167211809567495, −2.66907075111205383917085810778, −2.37938663941908880657930341002, −1.65730355152551961546014863016,
1.65730355152551961546014863016, 2.37938663941908880657930341002, 2.66907075111205383917085810778, 3.88261708380527167211809567495, 4.38492488245376512643072179509, 5.42927634164617286025699613855, 5.78640405170751959347105161981, 6.29205951881109080234144756635, 6.71778066430661336888582716183, 7.41739122861685143280509277723, 8.587851934459013834761684572790, 8.768961470526460026351817099329, 8.998106452521941953152706182178, 9.288995754727251390007447684528, 9.933228883600645345117312716392, 10.20704899738732125371168016271, 10.99769158155824883235202256723, 11.61430493196621781706902305185, 11.75485079284358130848965550474, 13.14580004509351118819641553580
