| L(s) = 1 | − 2·3-s + 10·4-s − 15·9-s − 20·12-s + 43·16-s + 46·25-s + 50·27-s − 150·36-s − 140·43-s − 86·48-s + 14·49-s + 280·61-s + 20·64-s − 92·75-s − 200·79-s + 140·81-s + 460·100-s + 500·108-s + 460·121-s + 127-s + 280·129-s + 131-s + 137-s + 139-s − 645·144-s − 28·147-s + 149-s + ⋯ |
| L(s) = 1 | − 2/3·3-s + 5/2·4-s − 5/3·9-s − 5/3·12-s + 2.68·16-s + 1.83·25-s + 1.85·27-s − 4.16·36-s − 3.25·43-s − 1.79·48-s + 2/7·49-s + 4.59·61-s + 5/16·64-s − 1.22·75-s − 2.53·79-s + 1.72·81-s + 23/5·100-s + 4.62·108-s + 3.80·121-s + 0.00787·127-s + 2.17·129-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 4.47·144-s − 0.190·147-s + 0.00671·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.586111389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.586111389\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{4} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 - 23 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - p T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 230 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 29 T + p^{2} T^{2} )^{2}( 1 + 29 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 302 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - 422 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 1502 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 2633 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 3350 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 35 T + p^{2} T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 1535 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 578 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 6530 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 70 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 3455 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 4462 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 50 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 2650 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 4310 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 18398 T^{2} + p^{4} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.30718738321887381547447337318, −7.27716898440537763414109432971, −7.21488432191410524703080186429, −6.82851408310983489560099241896, −6.70434769752232612602922111900, −6.45394541118912327145618877022, −6.35873592376760843750774166921, −5.92201156043958000081687636866, −5.79716698718108515766140041775, −5.52987907559908338322513476676, −5.23649576925270460072619655327, −5.02305846814481016646642361489, −4.71781964722330824873132535524, −4.52971770141197356109676394266, −3.81964372990801140886627928387, −3.62085620410495809970313397218, −3.39401653124591545206178056260, −2.81321770122396465563980074123, −2.79575816776815312544250655246, −2.60553141462978124827814970838, −2.20036027322627868719359080650, −1.72018070913872911810557135910, −1.49676342980095527406041375792, −0.834054630445934882341645898300, −0.36840503127744013091015699042,
0.36840503127744013091015699042, 0.834054630445934882341645898300, 1.49676342980095527406041375792, 1.72018070913872911810557135910, 2.20036027322627868719359080650, 2.60553141462978124827814970838, 2.79575816776815312544250655246, 2.81321770122396465563980074123, 3.39401653124591545206178056260, 3.62085620410495809970313397218, 3.81964372990801140886627928387, 4.52971770141197356109676394266, 4.71781964722330824873132535524, 5.02305846814481016646642361489, 5.23649576925270460072619655327, 5.52987907559908338322513476676, 5.79716698718108515766140041775, 5.92201156043958000081687636866, 6.35873592376760843750774166921, 6.45394541118912327145618877022, 6.70434769752232612602922111900, 6.82851408310983489560099241896, 7.21488432191410524703080186429, 7.27716898440537763414109432971, 7.30718738321887381547447337318
