| L(s) = 1 | + 6·3-s − 3·4-s − 16·7-s + 27·9-s − 18·12-s + 8·13-s − 7·16-s − 12·19-s − 96·21-s + 6·25-s + 108·27-s + 48·28-s − 52·31-s − 81·36-s + 60·37-s + 48·39-s + 84·43-s − 42·48-s + 94·49-s − 24·52-s − 72·57-s + 24·61-s − 432·63-s + 69·64-s + 4·67-s − 148·73-s + 36·75-s + ⋯ |
| L(s) = 1 | + 2·3-s − 3/4·4-s − 2.28·7-s + 3·9-s − 3/2·12-s + 8/13·13-s − 0.437·16-s − 0.631·19-s − 4.57·21-s + 6/25·25-s + 4·27-s + 12/7·28-s − 1.67·31-s − 9/4·36-s + 1.62·37-s + 1.23·39-s + 1.95·43-s − 7/8·48-s + 1.91·49-s − 0.461·52-s − 1.26·57-s + 0.393·61-s − 6.85·63-s + 1.07·64-s + 4/67·67-s − 2.02·73-s + 0.479·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.323172970\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.323172970\) |
| \(L(2)\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T^{2} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 6 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 402 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 1014 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 98 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 26 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 30 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3186 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 42 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 3018 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2054 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 2562 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 6518 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 74 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 12194 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 1586 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 62 T + p^{2} T^{2} )^{2} \) |
| 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
−16.39340489348794926799824590827, −16.01398418524725385160950626319, −15.78213262934442172356563398244, −14.71618178752052947563904800675, −14.56671741647811408569175428269, −13.56469871033383997019074328678, −13.34101850161211493925304591171, −12.77656968448705242970921055972, −12.57513390229127967014706317425, −11.00398082884895023266604412421, −10.07999574161445947999091575522, −9.625344277890424690454082788157, −9.026354364107742844429786346354, −8.782121035391266122095343451094, −7.68385747545354478235407501398, −6.93970335167476522991480177841, −6.11198907827616590669215168131, −4.29685542700832488628962029342, −3.62327587915032566364728411320, −2.68129674928404865106074845948,
2.68129674928404865106074845948, 3.62327587915032566364728411320, 4.29685542700832488628962029342, 6.11198907827616590669215168131, 6.93970335167476522991480177841, 7.68385747545354478235407501398, 8.782121035391266122095343451094, 9.026354364107742844429786346354, 9.625344277890424690454082788157, 10.07999574161445947999091575522, 11.00398082884895023266604412421, 12.57513390229127967014706317425, 12.77656968448705242970921055972, 13.34101850161211493925304591171, 13.56469871033383997019074328678, 14.56671741647811408569175428269, 14.71618178752052947563904800675, 15.78213262934442172356563398244, 16.01398418524725385160950626319, 16.39340489348794926799824590827
