| L(s) = 1 | − 6·9-s − 2·25-s − 32·31-s + 8·41-s + 36·49-s − 32·71-s + 32·79-s + 21·81-s + 40·89-s − 68·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 60·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 2·9-s − 2/5·25-s − 5.74·31-s + 1.24·41-s + 36/7·49-s − 3.79·71-s + 3.60·79-s + 7/3·81-s + 4.23·89-s − 6.18·121-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 + 4.61·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 3^{12} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 3^{12} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7679133855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7679133855\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T^{2} )^{6} \) |
| 5 | \( 1 + 2 T^{2} - 9 T^{4} - 196 T^{6} - 9 p^{2} T^{8} + 2 p^{4} T^{10} + p^{6} T^{12} \) |
| good | 7 | \( ( 1 - 18 T^{2} + 191 T^{4} - 1532 T^{6} + 191 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 11 | \( ( 1 + 34 T^{2} + 503 T^{4} + 5436 T^{6} + 503 p^{2} T^{8} + 34 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 13 | \( ( 1 - 30 T^{2} + 743 T^{4} - 10436 T^{6} + 743 p^{2} T^{8} - 30 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 17 | \( ( 1 - 66 T^{2} + 2255 T^{4} - 47324 T^{6} + 2255 p^{2} T^{8} - 66 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 19 | \( ( 1 + 54 T^{2} + 1367 T^{4} + 25652 T^{6} + 1367 p^{2} T^{8} + 54 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 23 | \( ( 1 - 2 p T^{2} + 1775 T^{4} - 40932 T^{6} + 1775 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} )^{2} \) |
| 29 | \( ( 1 + 66 T^{2} + 3207 T^{4} + 111228 T^{6} + 3207 p^{2} T^{8} + 66 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 31 | \( ( 1 + 8 T + 89 T^{2} + 432 T^{3} + 89 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 37 | \( ( 1 - 158 T^{2} + 11191 T^{4} - 496772 T^{6} + 11191 p^{2} T^{8} - 158 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 41 | \( ( 1 - 2 T + 23 T^{2} - 220 T^{3} + 23 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 43 | \( ( 1 - 130 T^{2} + 9815 T^{4} - 518268 T^{6} + 9815 p^{2} T^{8} - 130 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 47 | \( ( 1 - 222 T^{2} + 22367 T^{4} - 1328324 T^{6} + 22367 p^{2} T^{8} - 222 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 53 | \( ( 1 - 238 T^{2} + 26391 T^{4} - 1758052 T^{6} + 26391 p^{2} T^{8} - 238 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 59 | \( ( 1 + 178 T^{2} + 20567 T^{4} + 1418652 T^{6} + 20567 p^{2} T^{8} + 178 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 61 | \( ( 1 + 190 T^{2} + 20039 T^{4} + 1419204 T^{6} + 20039 p^{2} T^{8} + 190 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 67 | \( ( 1 - 274 T^{2} + 37127 T^{4} - 3112476 T^{6} + 37127 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 71 | \( ( 1 + 8 T + 133 T^{2} + 1008 T^{3} + 133 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 73 | \( ( 1 - 54 T^{2} + 2367 T^{4} - 531700 T^{6} + 2367 p^{2} T^{8} - 54 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 79 | \( ( 1 - 8 T + 233 T^{2} - 1200 T^{3} + 233 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 83 | \( ( 1 - 306 T^{2} + 50855 T^{4} - 5168732 T^{6} + 50855 p^{2} T^{8} - 306 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 89 | \( ( 1 - 10 T + 103 T^{2} - 396 T^{3} + 103 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 97 | \( ( 1 - 246 T^{2} + 39183 T^{4} - 4535476 T^{6} + 39183 p^{2} T^{8} - 246 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.41739657354067735958825228403, −2.39200131736231924995304642253, −2.39153944113779147208498354962, −2.32505348928801907646059480480, −2.16096308730562052727351758028, −2.14591509078287772576785165050, −2.13728478573134241164212952601, −2.08989489194908749608331015571, −1.89627863966205114191728670980, −1.85960592357421861478076960784, −1.65732293935353401716738906275, −1.59272425667077542372823537596, −1.49649088056203408669601335056, −1.47482349472830164618283012496, −1.23493145216200480700154482240, −1.13865178779676324125401420677, −1.09073304727314019575066454253, −0.975444950035737775336272719541, −0.898966212831908845903217499537, −0.810927044075421969783626520962, −0.60985970578617965420753781680, −0.37998610173395063241020068851, −0.28848559773233745626890572007, −0.21422113588713702318154523978, −0.084158929477579871299578668392,
0.084158929477579871299578668392, 0.21422113588713702318154523978, 0.28848559773233745626890572007, 0.37998610173395063241020068851, 0.60985970578617965420753781680, 0.810927044075421969783626520962, 0.898966212831908845903217499537, 0.975444950035737775336272719541, 1.09073304727314019575066454253, 1.13865178779676324125401420677, 1.23493145216200480700154482240, 1.47482349472830164618283012496, 1.49649088056203408669601335056, 1.59272425667077542372823537596, 1.65732293935353401716738906275, 1.85960592357421861478076960784, 1.89627863966205114191728670980, 2.08989489194908749608331015571, 2.13728478573134241164212952601, 2.14591509078287772576785165050, 2.16096308730562052727351758028, 2.32505348928801907646059480480, 2.39153944113779147208498354962, 2.39200131736231924995304642253, 2.41739657354067735958825228403
Plot not available for L-functions of degree greater than 10.
