| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s − 7-s + 2·8-s + 9-s − 2·12-s − 14-s + 2·16-s + 17-s + 18-s + 2·21-s − 4·24-s − 25-s + 2·27-s − 28-s + 2·32-s + 34-s + 36-s + 37-s + 2·42-s − 47-s − 4·48-s − 50-s − 2·51-s + 53-s + ⋯ |
| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s − 7-s + 2·8-s + 9-s − 2·12-s − 14-s + 2·16-s + 17-s + 18-s + 2·21-s − 4·24-s − 25-s + 2·27-s − 28-s + 2·32-s + 34-s + 36-s + 37-s + 2·42-s − 47-s − 4·48-s − 50-s − 2·51-s + 53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108241 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108241 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5982080586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5982080586\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 47 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 2 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 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
−11.83988006959751153266928827298, −11.61522390432652797060404954170, −11.34400716457985881643436985485, −10.90672704503313158336709174108, −10.16755714907701894495099559863, −10.16158649940227520378359334706, −9.688275521645396995568294994764, −8.624463388835655697389509257478, −8.169032658583608566069655164604, −7.46253857929727232494147872029, −6.95398221082477526294204008166, −6.62079931918439691961234634925, −5.92795201129495718593740288338, −5.63318403982791156057331750624, −5.40786190130025409057174137241, −4.54907603163398575805880359415, −4.19381485711651291004855249184, −3.29721858350856258410418911185, −2.63006638421219917034747835911, −1.31151289050695141982660107552,
1.31151289050695141982660107552, 2.63006638421219917034747835911, 3.29721858350856258410418911185, 4.19381485711651291004855249184, 4.54907603163398575805880359415, 5.40786190130025409057174137241, 5.63318403982791156057331750624, 5.92795201129495718593740288338, 6.62079931918439691961234634925, 6.95398221082477526294204008166, 7.46253857929727232494147872029, 8.169032658583608566069655164604, 8.624463388835655697389509257478, 9.688275521645396995568294994764, 10.16158649940227520378359334706, 10.16755714907701894495099559863, 10.90672704503313158336709174108, 11.34400716457985881643436985485, 11.61522390432652797060404954170, 11.83988006959751153266928827298
