| L(s) = 1 | + 2·7-s + 2·13-s + 25-s − 3·31-s + 37-s + 3·49-s + 61-s − 3·67-s + 2·73-s + 3·79-s + 4·91-s + 2·97-s − 3·103-s − 109-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 2·175-s + ⋯ |
| L(s) = 1 | + 2·7-s + 2·13-s + 25-s − 3·31-s + 37-s + 3·49-s + 61-s − 3·67-s + 2·73-s + 3·79-s + 4·91-s + 2·97-s − 3·103-s − 109-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 2·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.153414640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.153414640\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.866525362048845582821812445013, −8.820081740306154960590833537886, −8.263826248559455055939129373699, −8.059029675906616316052115662124, −7.63300297453339727998686477078, −7.39675079304989587397255431417, −6.79437257771431562415112274757, −6.53093312987643819916560418355, −5.91473502389928183082332716287, −5.64876366594014347599404886442, −5.23189641859203798204520284516, −4.96445288068687993402731273053, −4.28831107292738707892838601364, −4.14573848771854729491142315653, −3.45458522418613692084503487514, −3.34431945495599445606938385071, −2.20385379468149205852415791300, −2.18042506411695364806450510456, −1.24043189277363175749018302102, −1.18556202610363121711901065780,
1.18556202610363121711901065780, 1.24043189277363175749018302102, 2.18042506411695364806450510456, 2.20385379468149205852415791300, 3.34431945495599445606938385071, 3.45458522418613692084503487514, 4.14573848771854729491142315653, 4.28831107292738707892838601364, 4.96445288068687993402731273053, 5.23189641859203798204520284516, 5.64876366594014347599404886442, 5.91473502389928183082332716287, 6.53093312987643819916560418355, 6.79437257771431562415112274757, 7.39675079304989587397255431417, 7.63300297453339727998686477078, 8.059029675906616316052115662124, 8.263826248559455055939129373699, 8.820081740306154960590833537886, 8.866525362048845582821812445013
