| L(s) = 1 | − 2·2-s + 4-s − 7-s + 2·8-s − 2·9-s − 2·11-s + 2·14-s − 4·16-s + 4·18-s + 4·22-s + 2·23-s − 2·25-s − 28-s + 2·29-s + 2·32-s − 2·36-s − 2·43-s − 2·44-s − 4·46-s + 4·50-s − 2·53-s − 2·56-s − 4·58-s + 2·63-s + 3·64-s − 2·67-s − 4·72-s + ⋯ |
| L(s) = 1 | − 2·2-s + 4-s − 7-s + 2·8-s − 2·9-s − 2·11-s + 2·14-s − 4·16-s + 4·18-s + 4·22-s + 2·23-s − 2·25-s − 28-s + 2·29-s + 2·32-s − 2·36-s − 2·43-s − 2·44-s − 4·46-s + 4·50-s − 2·53-s − 2·56-s − 4·58-s + 2·63-s + 3·64-s − 2·67-s − 4·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10962721 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10962721 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01423599557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01423599557\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| 43 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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
−9.025039414325233654726678122075, −8.450361966855849921635811453407, −8.421049987936147605340650570703, −8.153226481502350323242030210615, −7.70828962419818778877108707674, −7.40114576199530353302863881532, −7.04995725516377102337934571241, −6.30979673092127123479491680445, −6.25166546449569114774525191881, −5.64391875104492880305826513093, −5.06326508530147604712346056837, −4.90452578923521867940498540895, −4.66959389478211386517829951693, −3.70543618305378118697274646827, −3.33328702464501934454540160028, −2.74862020248336242659602560803, −2.61376858094486118421661035279, −1.82637020642320401159290164248, −1.09774670390797591481437883832, −0.11450355215158241144927256593,
0.11450355215158241144927256593, 1.09774670390797591481437883832, 1.82637020642320401159290164248, 2.61376858094486118421661035279, 2.74862020248336242659602560803, 3.33328702464501934454540160028, 3.70543618305378118697274646827, 4.66959389478211386517829951693, 4.90452578923521867940498540895, 5.06326508530147604712346056837, 5.64391875104492880305826513093, 6.25166546449569114774525191881, 6.30979673092127123479491680445, 7.04995725516377102337934571241, 7.40114576199530353302863881532, 7.70828962419818778877108707674, 8.153226481502350323242030210615, 8.421049987936147605340650570703, 8.450361966855849921635811453407, 9.025039414325233654726678122075
