| L(s) = 1 | − 3-s − 2·4-s − 2·9-s + 2·12-s − 5·13-s − 3·23-s + 25-s + 5·27-s − 12·29-s + 4·36-s + 5·39-s − 11·43-s + 5·49-s + 10·52-s + 12·53-s − 13·61-s + 8·64-s + 3·69-s − 75-s − 7·79-s + 81-s + 12·87-s + 6·92-s − 2·100-s + 27·101-s − 20·103-s + 12·107-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 2/3·9-s + 0.577·12-s − 1.38·13-s − 0.625·23-s + 1/5·25-s + 0.962·27-s − 2.22·29-s + 2/3·36-s + 0.800·39-s − 1.67·43-s + 5/7·49-s + 1.38·52-s + 1.64·53-s − 1.66·61-s + 64-s + 0.361·69-s − 0.115·75-s − 0.787·79-s + 1/9·81-s + 1.28·87-s + 0.625·92-s − 1/5·100-s + 2.68·101-s − 1.97·103-s + 1.16·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 55 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 53 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 89 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 125 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 157 T^{2} + p^{2} T^{4} \) |
| 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
−11.00532084754246167124084101708, −10.31453851596519414293283116870, −9.865541943633370090286786133496, −9.259193378129729411314699302122, −8.838535630268282341179855629225, −8.222861338133290193246821662946, −7.47671691054370486042388375079, −6.99355924731825873030171106118, −6.06697710305086175522084812220, −5.48114874742225891130904665830, −4.95287287063942728295349208421, −4.29189245265897939806533711216, −3.38434697765581707429375510649, −2.21350441470184917104183116839, 0,
2.21350441470184917104183116839, 3.38434697765581707429375510649, 4.29189245265897939806533711216, 4.95287287063942728295349208421, 5.48114874742225891130904665830, 6.06697710305086175522084812220, 6.99355924731825873030171106118, 7.47671691054370486042388375079, 8.222861338133290193246821662946, 8.838535630268282341179855629225, 9.259193378129729411314699302122, 9.865541943633370090286786133496, 10.31453851596519414293283116870, 11.00532084754246167124084101708
