| L(s) = 1 | + 4-s − 5-s + 3·9-s − 4·11-s + 16-s + 12·19-s − 20-s − 4·25-s + 4·29-s + 8·31-s + 3·36-s − 4·44-s − 3·45-s − 13·49-s + 4·55-s − 20·59-s − 16·61-s + 64-s − 10·71-s + 12·76-s − 8·79-s − 80-s + 12·89-s − 12·95-s − 12·99-s − 4·100-s + 8·101-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.447·5-s + 9-s − 1.20·11-s + 1/4·16-s + 2.75·19-s − 0.223·20-s − 4/5·25-s + 0.742·29-s + 1.43·31-s + 1/2·36-s − 0.603·44-s − 0.447·45-s − 1.85·49-s + 0.539·55-s − 2.60·59-s − 2.04·61-s + 1/8·64-s − 1.18·71-s + 1.37·76-s − 0.900·79-s − 0.111·80-s + 1.27·89-s − 1.23·95-s − 1.20·99-s − 2/5·100-s + 0.796·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.207112567\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.207112567\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + T + p T^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 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
−11.16872163333009457316618804048, −10.28025762974353066563394220777, −10.09004531416049122481182188042, −9.580865457605610408706279565527, −8.851457547102649893241683814584, −7.948925922714141001807879270954, −7.53253451320559507885544451017, −7.45985560667588965978165906144, −6.41616277419392467413145284850, −5.88908450915949561478136129218, −4.92463422036437944045737589562, −4.61153453491182141362175201622, −3.35536338777490938542075146583, −2.88958900748636961412277796391, −1.44774618873205881453217331750,
1.44774618873205881453217331750, 2.88958900748636961412277796391, 3.35536338777490938542075146583, 4.61153453491182141362175201622, 4.92463422036437944045737589562, 5.88908450915949561478136129218, 6.41616277419392467413145284850, 7.45985560667588965978165906144, 7.53253451320559507885544451017, 7.948925922714141001807879270954, 8.851457547102649893241683814584, 9.580865457605610408706279565527, 10.09004531416049122481182188042, 10.28025762974353066563394220777, 11.16872163333009457316618804048
