| L(s) = 1 | + 2·2-s + 4-s + 2·7-s + 2·8-s + 4·14-s + 7·16-s − 6·17-s + 16·23-s + 27·25-s + 2·28-s + 32·31-s + 10·32-s − 12·34-s + 20·41-s + 32·46-s + 10·47-s − 25·49-s + 54·50-s + 4·56-s + 64·62-s + 13·64-s − 6·68-s + 18·71-s − 12·73-s − 12·79-s + 40·82-s − 16·89-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s + 0.755·7-s + 0.707·8-s + 1.06·14-s + 7/4·16-s − 1.45·17-s + 3.33·23-s + 27/5·25-s + 0.377·28-s + 5.74·31-s + 1.76·32-s − 2.05·34-s + 3.12·41-s + 4.71·46-s + 1.45·47-s − 3.57·49-s + 7.63·50-s + 0.534·56-s + 8.12·62-s + 13/8·64-s − 0.727·68-s + 2.13·71-s − 1.40·73-s − 1.35·79-s + 4.41·82-s − 1.69·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(28.33986587\) |
| \(L(\frac12)\) |
\(\approx\) |
\(28.33986587\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T + 3 T^{2} - 3 p T^{3} + 3 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \) |
| 3 | \( 1 \) |
| 13 | \( ( 1 + T^{2} )^{3} \) |
| good | 5 | \( ( 1 - 9 T^{2} + p^{2} T^{4} )^{3} \) |
| 7 | \( ( 1 - T + 2 p T^{2} - 15 T^{3} + 2 p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 11 | \( 1 - 26 T^{2} + 327 T^{4} - 3308 T^{6} + 327 p^{2} T^{8} - 26 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 + T + p T^{2} )^{6} \) |
| 19 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{3} \) |
| 23 | \( ( 1 - 8 T + 73 T^{2} - 336 T^{3} + 73 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( 1 - 38 T^{2} + 2487 T^{4} - 63572 T^{6} + 2487 p^{2} T^{8} - 38 p^{4} T^{10} + p^{6} T^{12} \) |
| 31 | \( ( 1 - 16 T + 161 T^{2} - 1056 T^{3} + 161 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 163 T^{2} + 12494 T^{4} - 579447 T^{6} + 12494 p^{2} T^{8} - 163 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 - 10 T + 91 T^{2} - 468 T^{3} + 91 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 211 T^{2} + 19666 T^{4} - 1070011 T^{6} + 19666 p^{2} T^{8} - 211 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( ( 1 - 5 T + 2 p T^{2} - 243 T^{3} + 2 p^{2} T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 53 | \( 1 - 214 T^{2} + 22023 T^{4} - 1425844 T^{6} + 22023 p^{2} T^{8} - 214 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( 1 - 94 T^{2} + 9671 T^{4} - 527844 T^{6} + 9671 p^{2} T^{8} - 94 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( 1 - 42 T^{2} + 9527 T^{4} - 255116 T^{6} + 9527 p^{2} T^{8} - 42 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( 1 - 182 T^{2} + 19111 T^{4} - 1505300 T^{6} + 19111 p^{2} T^{8} - 182 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 9 T + 182 T^{2} - 1007 T^{3} + 182 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( ( 1 + 6 T + 203 T^{2} + 812 T^{3} + 203 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( ( 1 + 6 T + 17 T^{2} - 500 T^{3} + 17 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 406 T^{2} + 74567 T^{4} - 7920084 T^{6} + 74567 p^{2} T^{8} - 406 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 8 T + 219 T^{2} + 1296 T^{3} + 219 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( ( 1 - 8 T + 183 T^{2} - 704 T^{3} + 183 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
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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
−5.26588350871209562906395654280, −5.06362553605733191639499257715, −4.83744953319062813119090709581, −4.78687350439005066273686782629, −4.61201029200319597303229312489, −4.60504532544078012141033954793, −4.48452113151079999515526324801, −4.35945084801611108103490773065, −4.19095662217913230636622011933, −4.06279804321927119811964906934, −3.54975930259906914426418039745, −3.40564069547304639830138829284, −3.21191133713263573914301049545, −2.93377107911527949477342034126, −2.90392742272844534258131750993, −2.82447640216236569165024588867, −2.77603752351118177811475109960, −2.37218407943372264496207390905, −2.16841775700042522476603607332, −1.94523706858942344942767110630, −1.32437778266372431537438098783, −1.16979071573393543096671472218, −0.956184802107055441323295599286, −0.947417399778584215624634021995, −0.75358876853674065754331025359,
0.75358876853674065754331025359, 0.947417399778584215624634021995, 0.956184802107055441323295599286, 1.16979071573393543096671472218, 1.32437778266372431537438098783, 1.94523706858942344942767110630, 2.16841775700042522476603607332, 2.37218407943372264496207390905, 2.77603752351118177811475109960, 2.82447640216236569165024588867, 2.90392742272844534258131750993, 2.93377107911527949477342034126, 3.21191133713263573914301049545, 3.40564069547304639830138829284, 3.54975930259906914426418039745, 4.06279804321927119811964906934, 4.19095662217913230636622011933, 4.35945084801611108103490773065, 4.48452113151079999515526324801, 4.60504532544078012141033954793, 4.61201029200319597303229312489, 4.78687350439005066273686782629, 4.83744953319062813119090709581, 5.06362553605733191639499257715, 5.26588350871209562906395654280
Plot not available for L-functions of degree greater than 10.
