| L(s) = 1 | + 2·2-s − 4-s − 8·8-s + 9-s + 11-s − 7·16-s + 2·18-s + 2·22-s − 5·23-s + 4·25-s + 11·29-s + 14·32-s − 36-s + 9·37-s + 10·43-s − 44-s − 10·46-s + 8·50-s − 6·53-s + 22·58-s + 35·64-s + 67-s + 4·71-s − 8·72-s + 18·74-s + 16·79-s − 8·81-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1/2·4-s − 2.82·8-s + 1/3·9-s + 0.301·11-s − 7/4·16-s + 0.471·18-s + 0.426·22-s − 1.04·23-s + 4/5·25-s + 2.04·29-s + 2.47·32-s − 1/6·36-s + 1.47·37-s + 1.52·43-s − 0.150·44-s − 1.47·46-s + 1.13·50-s − 0.824·53-s + 2.88·58-s + 35/8·64-s + 0.122·67-s + 0.474·71-s − 0.942·72-s + 2.09·74-s + 1.80·79-s − 8/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103243 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103243 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.032677315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.032677315\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 9 T + p T^{2} ) \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 36 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 108 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 51 T^{2} + p^{2} T^{4} \) |
| 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
−9.578214978324732639672195921030, −9.004091435082521609852171911359, −8.695837283864205248599805648506, −8.063920037283955064040691169883, −7.67231731041940318174121782766, −6.67672647471498942102283360255, −6.30322321088606872928877489120, −5.91472830926969588487263306404, −5.17957234254618180284917566997, −4.76306582814486018866544798170, −4.24873773635288204653709517801, −3.88470819087452431807723266696, −3.08674579531974236386492319635, −2.51748493458935850834729218512, −0.880941825614909303846840296459,
0.880941825614909303846840296459, 2.51748493458935850834729218512, 3.08674579531974236386492319635, 3.88470819087452431807723266696, 4.24873773635288204653709517801, 4.76306582814486018866544798170, 5.17957234254618180284917566997, 5.91472830926969588487263306404, 6.30322321088606872928877489120, 6.67672647471498942102283360255, 7.67231731041940318174121782766, 8.063920037283955064040691169883, 8.695837283864205248599805648506, 9.004091435082521609852171911359, 9.578214978324732639672195921030
