| L(s) = 1 | − 2-s − 5-s + 8-s + 3·9-s + 10-s − 4·11-s + 12·13-s − 16-s + 2·17-s − 3·18-s + 4·22-s − 12·26-s + 12·29-s + 8·31-s − 2·34-s + 10·37-s − 40-s − 4·41-s + 8·43-s − 3·45-s + 8·47-s + 2·53-s + 4·55-s − 12·58-s − 8·59-s − 14·61-s − 8·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.447·5-s + 0.353·8-s + 9-s + 0.316·10-s − 1.20·11-s + 3.32·13-s − 1/4·16-s + 0.485·17-s − 0.707·18-s + 0.852·22-s − 2.35·26-s + 2.22·29-s + 1.43·31-s − 0.342·34-s + 1.64·37-s − 0.158·40-s − 0.624·41-s + 1.21·43-s − 0.447·45-s + 1.16·47-s + 0.274·53-s + 0.539·55-s − 1.57·58-s − 1.04·59-s − 1.79·61-s − 1.01·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.403800546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.403800546\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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
−10.84055637465955460928553764854, −10.76139131892337034215290023987, −10.21550282238924816162189280291, −10.13489758514894369194710698770, −9.273618253358299557322270729048, −8.890425400856893629885893266498, −8.431652828118773300063315476581, −8.159904773946499751384773782109, −7.68069825620743439231099936605, −7.33803698449132475482490186730, −6.34215233745206491832720233826, −6.33986460037127327226140026893, −5.75040355798917216948881540916, −4.93459576263490344623743368010, −4.18418482461229828180603917816, −4.16244171021988846157080560592, −3.16563083441900853027794010464, −2.69194193851695474860010368735, −1.31546678517991833851215261671, −1.02043612726503043721379502445,
1.02043612726503043721379502445, 1.31546678517991833851215261671, 2.69194193851695474860010368735, 3.16563083441900853027794010464, 4.16244171021988846157080560592, 4.18418482461229828180603917816, 4.93459576263490344623743368010, 5.75040355798917216948881540916, 6.33986460037127327226140026893, 6.34215233745206491832720233826, 7.33803698449132475482490186730, 7.68069825620743439231099936605, 8.159904773946499751384773782109, 8.431652828118773300063315476581, 8.890425400856893629885893266498, 9.273618253358299557322270729048, 10.13489758514894369194710698770, 10.21550282238924816162189280291, 10.76139131892337034215290023987, 10.84055637465955460928553764854
