| L(s) = 1 | − 8·4-s + 6·7-s + 12·11-s + 36·16-s − 20·23-s + 26·25-s − 48·28-s − 4·43-s − 96·44-s + 23·49-s − 120·64-s + 72·77-s + 160·92-s − 208·100-s + 216·112-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 120·161-s + 163-s + 167-s − 22·169-s + ⋯ |
| L(s) = 1 | − 4·4-s + 2.26·7-s + 3.61·11-s + 9·16-s − 4.17·23-s + 26/5·25-s − 9.07·28-s − 0.609·43-s − 14.4·44-s + 23/7·49-s − 15·64-s + 8.20·77-s + 16.6·92-s − 20.7·100-s + 20.4·112-s + 6/11·121-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 − 9.45·161-s + 0.0783·163-s + 0.0773·167-s − 1.69·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 7^{16} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 7^{16} \cdot 19^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4441661349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4441661349\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{8} \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 - 3 T + 2 T^{2} - p T^{3} + 26 T^{4} - p^{2} T^{5} + 2 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 + 6 T^{2} + 640 T^{4} + 3882 T^{6} + 227102 T^{8} + 3882 p^{2} T^{10} + 640 p^{4} T^{12} + 6 p^{6} T^{14} + p^{8} T^{16} \) |
| good | 5 | \( ( 1 - 13 T^{2} + 137 T^{4} - 973 T^{6} + 5528 T^{8} - 973 p^{2} T^{10} + 137 p^{4} T^{12} - 13 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 11 | \( ( 1 - 3 T + 21 T^{2} - 29 T^{3} + 172 T^{4} - 29 p T^{5} + 21 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 13 | \( ( 1 + 11 T^{2} + 692 T^{4} + 5485 T^{6} + 176694 T^{8} + 5485 p^{2} T^{10} + 692 p^{4} T^{12} + 11 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 17 | \( ( 1 - 5 p T^{2} + 3674 T^{4} - 103723 T^{6} + 2072330 T^{8} - 103723 p^{2} T^{10} + 3674 p^{4} T^{12} - 5 p^{7} T^{14} + p^{8} T^{16} )^{2} \) |
| 23 | \( ( 1 + 5 T + 72 T^{2} + 293 T^{3} + 2350 T^{4} + 293 p T^{5} + 72 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 29 | \( ( 1 - 16 T^{2} + 2264 T^{4} - 31119 T^{6} + 2425240 T^{8} - 31119 p^{2} T^{10} + 2264 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 31 | \( ( 1 + 176 T^{2} + 14924 T^{4} + 797392 T^{6} + 29398374 T^{8} + 797392 p^{2} T^{10} + 14924 p^{4} T^{12} + 176 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 37 | \( ( 1 - 157 T^{2} + 11195 T^{4} - 504471 T^{6} + 18847328 T^{8} - 504471 p^{2} T^{10} + 11195 p^{4} T^{12} - 157 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 41 | \( ( 1 + 143 T^{2} + 8571 T^{4} + 285389 T^{6} + 8687552 T^{8} + 285389 p^{2} T^{10} + 8571 p^{4} T^{12} + 143 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 43 | \( ( 1 + T + 115 T^{2} - 25 T^{3} + 6320 T^{4} - 25 p T^{5} + 115 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 47 | \( ( 1 - 123 T^{2} + 10099 T^{4} - 526357 T^{6} + 26613240 T^{8} - 526357 p^{2} T^{10} + 10099 p^{4} T^{12} - 123 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 53 | \( ( 1 - 248 T^{2} + 26360 T^{4} - 1682079 T^{6} + 88298168 T^{8} - 1682079 p^{2} T^{10} + 26360 p^{4} T^{12} - 248 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 59 | \( ( 1 + 402 T^{2} + 74292 T^{4} + 8211335 T^{6} + 592643070 T^{8} + 8211335 p^{2} T^{10} + 74292 p^{4} T^{12} + 402 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 61 | \( ( 1 - 177 T^{2} + 19669 T^{4} - 1635501 T^{6} + 108694016 T^{8} - 1635501 p^{2} T^{10} + 19669 p^{4} T^{12} - 177 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 67 | \( ( 1 - 267 T^{2} + 33950 T^{4} - 2958861 T^{6} + 212072418 T^{8} - 2958861 p^{2} T^{10} + 33950 p^{4} T^{12} - 267 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 71 | \( ( 1 - 269 T^{2} + 40127 T^{4} - 4333599 T^{6} + 354617000 T^{8} - 4333599 p^{2} T^{10} + 40127 p^{4} T^{12} - 269 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 73 | \( ( 1 - 189 T^{2} + 22890 T^{4} - 2313667 T^{6} + 194702538 T^{8} - 2313667 p^{2} T^{10} + 22890 p^{4} T^{12} - 189 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 79 | \( ( 1 - 385 T^{2} + 77591 T^{4} - 10205739 T^{6} + 951064184 T^{8} - 10205739 p^{2} T^{10} + 77591 p^{4} T^{12} - 385 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 83 | \( ( 1 - 426 T^{2} + 80704 T^{4} - 9483238 T^{6} + 854616990 T^{8} - 9483238 p^{2} T^{10} + 80704 p^{4} T^{12} - 426 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 89 | \( ( 1 + 347 T^{2} + 57807 T^{4} + 6248621 T^{6} + 566720048 T^{8} + 6248621 p^{2} T^{10} + 57807 p^{4} T^{12} + 347 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 97 | \( ( 1 + 415 T^{2} + 91843 T^{4} + 14005949 T^{6} + 1576870832 T^{8} + 14005949 p^{2} T^{10} + 91843 p^{4} T^{12} + 415 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.14747877858586417954991478588, −2.12630131515660154972018074459, −2.12202787722154549770945294503, −2.11273199845187998479557587924, −1.88824278322743020140997897548, −1.60738367840381044228248959346, −1.59750569909296264189942503255, −1.56656875875285288556257802293, −1.51960569644130756322176757747, −1.48414405510580307351023276600, −1.45014489870084292666163518882, −1.31144787958443501829381180419, −1.28075015038763151820501366069, −1.22654358852326577417090405808, −1.17964508129986407415835960311, −1.08214825691079690004123870952, −0.999311627033494224895790897953, −0.977530807949581123951408510949, −0.71618471907976159812304710205, −0.68948031058071985657105323218, −0.63526401000962373230934313769, −0.40547963138297813420673561325, −0.22530236566917116635404377149, −0.18300487922820372965487698709, −0.05572776636192827014778208392,
0.05572776636192827014778208392, 0.18300487922820372965487698709, 0.22530236566917116635404377149, 0.40547963138297813420673561325, 0.63526401000962373230934313769, 0.68948031058071985657105323218, 0.71618471907976159812304710205, 0.977530807949581123951408510949, 0.999311627033494224895790897953, 1.08214825691079690004123870952, 1.17964508129986407415835960311, 1.22654358852326577417090405808, 1.28075015038763151820501366069, 1.31144787958443501829381180419, 1.45014489870084292666163518882, 1.48414405510580307351023276600, 1.51960569644130756322176757747, 1.56656875875285288556257802293, 1.59750569909296264189942503255, 1.60738367840381044228248959346, 1.88824278322743020140997897548, 2.11273199845187998479557587924, 2.12202787722154549770945294503, 2.12630131515660154972018074459, 2.14747877858586417954991478588
Plot not available for L-functions of degree greater than 10.
