| L(s) = 1 | − 2·2-s + 3·4-s + 7-s − 4·8-s − 2·11-s + 7·13-s − 2·14-s + 5·16-s − 8·17-s − 19-s + 4·22-s + 8·23-s + 5·25-s − 14·26-s + 3·28-s + 6·29-s − 6·32-s + 16·34-s − 22·37-s + 2·38-s − 2·41-s + 43-s − 6·44-s − 16·46-s − 6·49-s − 10·50-s + 21·52-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.377·7-s − 1.41·8-s − 0.603·11-s + 1.94·13-s − 0.534·14-s + 5/4·16-s − 1.94·17-s − 0.229·19-s + 0.852·22-s + 1.66·23-s + 25-s − 2.74·26-s + 0.566·28-s + 1.11·29-s − 1.06·32-s + 2.74·34-s − 3.61·37-s + 0.324·38-s − 0.312·41-s + 0.152·43-s − 0.904·44-s − 2.35·46-s − 6/7·49-s − 1.41·50-s + 2.91·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9311438944\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9311438944\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 9 T - 16 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 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
−9.322747020946160603025539826405, −8.963895827119563481644785496842, −8.836989766242702422706390121396, −8.467479736058126633470448705806, −8.310239821619570720046613032990, −7.68953111332205773162271046527, −7.19540356045492041852148375408, −6.75148099779236910569930445893, −6.59933708802577855153783765749, −6.25233373999320327905126107921, −5.55729697749046556000013235355, −4.98130421848402284513316739777, −4.83667711201413920896332494834, −4.02974237278753007064491355329, −3.43945692590064679379958635736, −3.02478152442257078750273375010, −2.45896161537724230350925291932, −1.70359589420274745261806602474, −1.38749117939702852151404167071, −0.48859868605872167485237745520,
0.48859868605872167485237745520, 1.38749117939702852151404167071, 1.70359589420274745261806602474, 2.45896161537724230350925291932, 3.02478152442257078750273375010, 3.43945692590064679379958635736, 4.02974237278753007064491355329, 4.83667711201413920896332494834, 4.98130421848402284513316739777, 5.55729697749046556000013235355, 6.25233373999320327905126107921, 6.59933708802577855153783765749, 6.75148099779236910569930445893, 7.19540356045492041852148375408, 7.68953111332205773162271046527, 8.310239821619570720046613032990, 8.467479736058126633470448705806, 8.836989766242702422706390121396, 8.963895827119563481644785496842, 9.322747020946160603025539826405
