L(s) = 1 | + 3·2-s + 3·4-s − 2·8-s − 3·13-s − 9·16-s − 9·26-s + 3·31-s − 9·32-s − 9·52-s + 9·62-s + 3·64-s + 6·104-s − 12·107-s − 3·113-s − 3·121-s + 9·124-s + 2·125-s + 127-s + 18·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
L(s) = 1 | + 3·2-s + 3·4-s − 2·8-s − 3·13-s − 9·16-s − 9·26-s + 3·31-s − 9·32-s − 9·52-s + 9·62-s + 3·64-s + 6·104-s − 12·107-s − 3·113-s − 3·121-s + 9·124-s + 2·125-s + 127-s + 18·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{18} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{18} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1562342040\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1562342040\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( ( 1 - T + T^{2} )^{3} \) |
| 3 | \( 1 \) |
| 13 | \( ( 1 + T + T^{2} )^{3} \) |
good | 5 | \( ( 1 - T^{3} + T^{6} )^{2} \) |
| 7 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 11 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 17 | \( ( 1 - T^{3} + T^{6} )^{2} \) |
| 19 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 23 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 29 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 31 | \( ( 1 - T )^{6}( 1 + T + T^{2} )^{3} \) |
| 37 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 41 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 43 | \( ( 1 + T^{3} + T^{6} )^{2} \) |
| 47 | \( ( 1 - T^{3} + T^{6} )^{2} \) |
| 53 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 59 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 61 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 67 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 71 | \( ( 1 - T^{3} + T^{6} )^{2} \) |
| 73 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 79 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 83 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
| 89 | \( ( 1 - T )^{6}( 1 + T )^{6} \) |
| 97 | \( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.69239170164381780224682960116, −4.62159580546650659645236845382, −4.55237577424930014543191498964, −4.48762252774925482148859025575, −4.11736787987896011511390806745, −3.95956553388492202963162497915, −3.94078217396504192312297011477, −3.91216246553446994489744036426, −3.76700983914554452879312746881, −3.52888488343768563498869120316, −3.47914214937857586765163882822, −2.93528081555984216211104886028, −2.83908497302343051494416634656, −2.75557608694147840339422236317, −2.74442327573442324846179389723, −2.69058569325040982868417568263, −2.54509936796200466928039440813, −2.52771322891463967329343798997, −1.98950187653924777123905140316, −1.95686677742981913589531432341, −1.51662194028972419632152534636, −1.32895126404561611978888288606, −1.11340207452450151835295177409, −0.78353535400114509445401010587, −0.083638204825075081030009234183,
0.083638204825075081030009234183, 0.78353535400114509445401010587, 1.11340207452450151835295177409, 1.32895126404561611978888288606, 1.51662194028972419632152534636, 1.95686677742981913589531432341, 1.98950187653924777123905140316, 2.52771322891463967329343798997, 2.54509936796200466928039440813, 2.69058569325040982868417568263, 2.74442327573442324846179389723, 2.75557608694147840339422236317, 2.83908497302343051494416634656, 2.93528081555984216211104886028, 3.47914214937857586765163882822, 3.52888488343768563498869120316, 3.76700983914554452879312746881, 3.91216246553446994489744036426, 3.94078217396504192312297011477, 3.95956553388492202963162497915, 4.11736787987896011511390806745, 4.48762252774925482148859025575, 4.55237577424930014543191498964, 4.62159580546650659645236845382, 4.69239170164381780224682960116
Plot not available for L-functions of degree greater than 10.