| L(s) = 1 | − 3-s + 3·4-s − 7-s − 3·12-s − 13-s + 6·16-s − 17-s + 21-s − 23-s + 3·25-s − 3·28-s + 39-s − 41-s − 43-s − 47-s − 6·48-s + 51-s − 3·52-s − 53-s − 59-s + 10·64-s − 3·68-s + 69-s − 71-s − 3·75-s − 83-s + 3·84-s + ⋯ |
| L(s) = 1 | − 3-s + 3·4-s − 7-s − 3·12-s − 13-s + 6·16-s − 17-s + 21-s − 23-s + 3·25-s − 3·28-s + 39-s − 41-s − 43-s − 47-s − 6·48-s + 51-s − 3·52-s − 53-s − 59-s + 10·64-s − 3·68-s + 69-s − 71-s − 3·75-s − 83-s + 3·84-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(467^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(467^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7571000367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7571000367\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 467 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 3 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 7 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 13 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 17 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 23 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 41 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 43 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 47 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 53 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 59 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 71 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 83 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 89 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 97 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.23433303295823125467383245330, −9.893521161743215929482898759268, −9.723931792593346066749891930352, −9.354511904140891677412402739686, −8.718642439284746687323609908428, −8.497611884057402239401196582957, −8.166577805712224433018579760114, −7.73138904662540327436911629105, −7.48097308862250021217943065878, −6.98715100282491274824774559738, −6.77532632542757166291585652451, −6.63549176413813228357902532529, −6.60888335051796937011489608478, −5.95289743880671994896173238330, −5.78451008545206650485783450356, −5.54095459843366247772041767344, −4.83509476189974862396029463534, −4.81228629461629871289296641376, −4.07154303783045488231560221465, −3.33739813868354473654185232548, −3.04784260920514135463153142273, −2.95031351893988248038314420176, −2.37737780748307281671064762218, −1.84652980298499369410297571995, −1.38831536471378564959270231894,
1.38831536471378564959270231894, 1.84652980298499369410297571995, 2.37737780748307281671064762218, 2.95031351893988248038314420176, 3.04784260920514135463153142273, 3.33739813868354473654185232548, 4.07154303783045488231560221465, 4.81228629461629871289296641376, 4.83509476189974862396029463534, 5.54095459843366247772041767344, 5.78451008545206650485783450356, 5.95289743880671994896173238330, 6.60888335051796937011489608478, 6.63549176413813228357902532529, 6.77532632542757166291585652451, 6.98715100282491274824774559738, 7.48097308862250021217943065878, 7.73138904662540327436911629105, 8.166577805712224433018579760114, 8.497611884057402239401196582957, 8.718642439284746687323609908428, 9.354511904140891677412402739686, 9.723931792593346066749891930352, 9.893521161743215929482898759268, 10.23433303295823125467383245330
