| L(s) = 1 | − 16·2-s + 136·4-s − 816·8-s + 3.87e3·16-s − 16·23-s + 6·25-s − 1.55e4·32-s + 256·46-s − 96·50-s + 32·53-s + 5.42e4·64-s − 8·79-s − 2.17e3·92-s + 816·100-s − 512·106-s + 72·107-s + 32·109-s − 48·113-s + 76·121-s + 127-s − 1.70e5·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 11.3·2-s + 68·4-s − 288.·8-s + 969·16-s − 3.33·23-s + 6/5·25-s − 2.74e3·32-s + 37.7·46-s − 13.5·50-s + 4.39·53-s + 6.78e3·64-s − 0.900·79-s − 226.·92-s + 81.5·100-s − 49.7·106-s + 6.96·107-s + 3.06·109-s − 4.51·113-s + 6.90·121-s + 0.0887·127-s − 1.50e4·128-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{16} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{16} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0004307786115\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0004307786115\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T )^{16} \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 6 T^{2} + 17 T^{4} + 42 T^{6} - 876 T^{8} + 42 p^{2} T^{10} + 17 p^{4} T^{12} - 6 p^{6} T^{14} + p^{8} T^{16} \) |
| 7 | \( 1 \) |
| good | 11 | \( ( 1 - 38 T^{2} + 514 T^{4} - 1376 T^{6} - 21281 T^{8} - 1376 p^{2} T^{10} + 514 p^{4} T^{12} - 38 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 13 | \( ( 1 + 32 T^{2} + 473 T^{4} + 8616 T^{6} + 148784 T^{8} + 8616 p^{2} T^{10} + 473 p^{4} T^{12} + 32 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 17 | \( ( 1 - 52 T^{2} + 1784 T^{4} - 42108 T^{6} + 805934 T^{8} - 42108 p^{2} T^{10} + 1784 p^{4} T^{12} - 52 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 19 | \( ( 1 - 42 T^{2} + 1613 T^{4} - 45630 T^{6} + 903912 T^{8} - 45630 p^{2} T^{10} + 1613 p^{4} T^{12} - 42 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 23 | \( ( 1 + 4 T + 39 T^{2} + 12 T^{3} + 442 T^{4} + 12 p T^{5} + 39 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 29 | \( ( 1 - 110 T^{2} + 5617 T^{4} - 185030 T^{6} + 5268820 T^{8} - 185030 p^{2} T^{10} + 5617 p^{4} T^{12} - 110 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 31 | \( ( 1 - 6 T^{2} + 1673 T^{4} - 1134 T^{6} + 1898388 T^{8} - 1134 p^{2} T^{10} + 1673 p^{4} T^{12} - 6 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 37 | \( ( 1 - 184 T^{2} + 15497 T^{4} - 826872 T^{6} + 33651488 T^{8} - 826872 p^{2} T^{10} + 15497 p^{4} T^{12} - 184 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 41 | \( ( 1 + 84 T^{2} + 9077 T^{4} + 448128 T^{6} + 24944604 T^{8} + 448128 p^{2} T^{10} + 9077 p^{4} T^{12} + 84 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 43 | \( ( 1 - 196 T^{2} + 20648 T^{4} - 1438284 T^{6} + 72218798 T^{8} - 1438284 p^{2} T^{10} + 20648 p^{4} T^{12} - 196 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 47 | \( ( 1 - 270 T^{2} + 34853 T^{4} - 2827818 T^{6} + 158149800 T^{8} - 2827818 p^{2} T^{10} + 34853 p^{4} T^{12} - 270 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 53 | \( ( 1 - 8 T + 168 T^{2} - 1044 T^{3} + 13021 T^{4} - 1044 p T^{5} + 168 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 59 | \( ( 1 + 426 T^{2} + 81485 T^{4} + 9169758 T^{6} + 665481384 T^{8} + 9169758 p^{2} T^{10} + 81485 p^{4} T^{12} + 426 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 61 | \( ( 1 - 68 T^{2} + 13144 T^{4} - 707276 T^{6} + 70076494 T^{8} - 707276 p^{2} T^{10} + 13144 p^{4} T^{12} - 68 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 67 | \( ( 1 - 352 T^{2} + 61868 T^{4} - 7006944 T^{6} + 554935238 T^{8} - 7006944 p^{2} T^{10} + 61868 p^{4} T^{12} - 352 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 71 | \( ( 1 + 52 T^{2} + 10456 T^{4} + 567388 T^{6} + 78505582 T^{8} + 567388 p^{2} T^{10} + 10456 p^{4} T^{12} + 52 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 73 | \( ( 1 + 312 T^{2} + 51644 T^{4} + 5812104 T^{6} + 484685382 T^{8} + 5812104 p^{2} T^{10} + 51644 p^{4} T^{12} + 312 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 79 | \( ( 1 + 2 T + 59 T^{2} + 870 T^{3} + 6116 T^{4} + 870 p T^{5} + 59 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 83 | \( ( 1 - 250 T^{2} + 32249 T^{4} - 3769146 T^{6} + 372693188 T^{8} - 3769146 p^{2} T^{10} + 32249 p^{4} T^{12} - 250 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 89 | \( ( 1 + 492 T^{2} + 112712 T^{4} + 16213956 T^{6} + 1669259214 T^{8} + 16213956 p^{2} T^{10} + 112712 p^{4} T^{12} + 492 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 97 | \( ( 1 + 438 T^{2} + 104849 T^{4} + 16454334 T^{6} + 1868658564 T^{8} + 16454334 p^{2} T^{10} + 104849 p^{4} T^{12} + 438 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
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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
−1.96042199629122496061609465574, −1.90889492896928326687120216250, −1.60929851844538661469353290146, −1.56960403191792476739650508836, −1.52479183390533444269040880724, −1.49467429421429288332896521971, −1.46654511804521036706397097683, −1.42669566128243200415796661143, −1.40892753140194695367390469377, −1.29673352146038408363259837430, −1.28925041496334317884997137308, −1.21972795199626931034481142081, −1.03485238732926539047806962317, −0.984852313685248351949572905090, −0.883662549777685046837448175147, −0.75738401136725853509501513011, −0.70896788283716652567581280321, −0.69146989136365419914772884530, −0.54343483280042718524010465337, −0.46625145767607682321401642447, −0.43185993766307105973892399910, −0.40688400338634820258337564739, −0.40425424988187079776725570228, −0.091693106533904806507233147203, −0.01908769349014892673757911074,
0.01908769349014892673757911074, 0.091693106533904806507233147203, 0.40425424988187079776725570228, 0.40688400338634820258337564739, 0.43185993766307105973892399910, 0.46625145767607682321401642447, 0.54343483280042718524010465337, 0.69146989136365419914772884530, 0.70896788283716652567581280321, 0.75738401136725853509501513011, 0.883662549777685046837448175147, 0.984852313685248351949572905090, 1.03485238732926539047806962317, 1.21972795199626931034481142081, 1.28925041496334317884997137308, 1.29673352146038408363259837430, 1.40892753140194695367390469377, 1.42669566128243200415796661143, 1.46654511804521036706397097683, 1.49467429421429288332896521971, 1.52479183390533444269040880724, 1.56960403191792476739650508836, 1.60929851844538661469353290146, 1.90889492896928326687120216250, 1.96042199629122496061609465574
Plot not available for L-functions of degree greater than 10.
