| L(s) = 1 | + 12·9-s + 14·19-s + 10·29-s + 10·31-s + 28·49-s + 16·59-s − 6·61-s + 6·71-s + 18·79-s + 76·81-s − 12·89-s − 26·101-s + 18·109-s − 24·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 33·169-s + 168·171-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 4·9-s + 3.21·19-s + 1.85·29-s + 1.79·31-s + 4·49-s + 2.08·59-s − 0.768·61-s + 0.712·71-s + 2.02·79-s + 76/9·81-s − 1.27·89-s − 2.58·101-s + 1.72·109-s − 2.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.53·169-s + 12.8·171-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12} \cdot 61^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{12} \cdot 61^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(55.59324538\) |
| \(L(\frac12)\) |
\(\approx\) |
\(55.59324538\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 61 | \( ( 1 + T )^{6} \) |
| good | 3 | \( 1 - 4 p T^{2} + 68 T^{4} - 245 T^{6} + 68 p^{2} T^{8} - 4 p^{5} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 4 p T^{2} + 8 p^{2} T^{4} - 69 p^{2} T^{6} + 8 p^{4} T^{8} - 4 p^{5} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 + 12 T^{2} + 7 T^{3} + 12 p T^{4} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 33 T^{2} + 737 T^{4} - 10721 T^{6} + 737 p^{2} T^{8} - 33 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 64 T^{2} + 2132 T^{4} - 44481 T^{6} + 2132 p^{2} T^{8} - 64 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 - 7 T + 43 T^{2} - 259 T^{3} + 43 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 41 T^{2} - 123 T^{4} + 26551 T^{6} - 123 p^{2} T^{8} - 41 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 5 T + 65 T^{2} - 193 T^{3} + 65 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 - 5 T + 99 T^{2} - 311 T^{3} + 99 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 61 T^{2} + 1117 T^{4} - 8665 T^{6} + 1117 p^{2} T^{8} - 61 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 60 T^{2} - 189 T^{3} + 60 p T^{4} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 225 T^{2} + 22373 T^{4} - 1250201 T^{6} + 22373 p^{2} T^{8} - 225 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 181 T^{2} + 16257 T^{4} - 936145 T^{6} + 16257 p^{2} T^{8} - 181 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 229 T^{2} + 24113 T^{4} - 1566369 T^{6} + 24113 p^{2} T^{8} - 229 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 - 8 T + 154 T^{2} - 957 T^{3} + 154 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 257 T^{2} + 33577 T^{4} - 2791817 T^{6} + 33577 p^{2} T^{8} - 257 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 3 T + 188 T^{2} - 455 T^{3} + 188 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 4 p T^{2} + 41704 T^{4} - 3752833 T^{6} + 41704 p^{2} T^{8} - 4 p^{5} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 9 T + 173 T^{2} - 889 T^{3} + 173 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 180 T^{2} + 24824 T^{4} - 2500229 T^{6} + 24824 p^{2} T^{8} - 180 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 6 T + 188 T^{2} + 1181 T^{3} + 188 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 184 T^{2} + 13652 T^{4} - 650121 T^{6} + 13652 p^{2} T^{8} - 184 p^{4} T^{10} + p^{6} T^{12} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.21947798923534413943252980196, −3.97091197713339769577098824726, −3.92474046270551950558564869917, −3.79567221090546307065670448381, −3.58819694271946620508173974135, −3.39252568579230120520503456507, −3.37982012991687136917209492115, −3.22367773784186976538032576290, −2.92651565118919353813184890188, −2.76373725436746039673543759562, −2.75300741906320418563513910898, −2.65121677160228175788221265588, −2.40033270945943512103722649557, −2.26439266185041340849671683915, −2.07901238521038487037891540010, −1.74489505468882917110123448034, −1.73770386437115605411243788054, −1.59692624826520438389687353523, −1.41983517819124762612709122588, −1.20344309057368596882594297152, −0.915578589348141074357883362952, −0.836124331022996148307494562378, −0.803320135064208524705550496560, −0.72071060885333101483281213575, −0.40610729356868451673432032711,
0.40610729356868451673432032711, 0.72071060885333101483281213575, 0.803320135064208524705550496560, 0.836124331022996148307494562378, 0.915578589348141074357883362952, 1.20344309057368596882594297152, 1.41983517819124762612709122588, 1.59692624826520438389687353523, 1.73770386437115605411243788054, 1.74489505468882917110123448034, 2.07901238521038487037891540010, 2.26439266185041340849671683915, 2.40033270945943512103722649557, 2.65121677160228175788221265588, 2.75300741906320418563513910898, 2.76373725436746039673543759562, 2.92651565118919353813184890188, 3.22367773784186976538032576290, 3.37982012991687136917209492115, 3.39252568579230120520503456507, 3.58819694271946620508173974135, 3.79567221090546307065670448381, 3.92474046270551950558564869917, 3.97091197713339769577098824726, 4.21947798923534413943252980196
Plot not available for L-functions of degree greater than 10.
