L(s) = 1 | − 12·9-s + 16·23-s + 136·25-s + 80·29-s + 128·37-s + 112·43-s − 144·53-s + 64·67-s − 224·71-s + 432·79-s + 90·81-s − 656·107-s + 16·109-s + 512·113-s − 720·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 640·169-s + 173-s + 179-s + ⋯ |
L(s) = 1 | − 4/3·9-s + 0.695·23-s + 5.43·25-s + 2.75·29-s + 3.45·37-s + 2.60·43-s − 2.71·53-s + 0.955·67-s − 3.15·71-s + 5.46·79-s + 10/9·81-s − 6.13·107-s + 0.146·109-s + 4.53·113-s − 5.95·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 3.78·169-s + 0.00578·173-s + 0.00558·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(33.97250291\) |
\(L(\frac12)\) |
\(\approx\) |
\(33.97250291\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( ( 1 + p T^{2} )^{4} \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 136 T^{2} + 8952 T^{4} - 379112 T^{6} + 11234546 T^{8} - 379112 p^{4} T^{10} + 8952 p^{8} T^{12} - 136 p^{12} T^{14} + p^{16} T^{16} \) |
| 11 | \( ( 1 + 360 T^{2} + 48 T^{3} + 61130 T^{4} + 48 p^{2} T^{5} + 360 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( 1 - 640 T^{2} + 201024 T^{4} - 3606656 p T^{6} + 8914586690 T^{8} - 3606656 p^{5} T^{10} + 201024 p^{8} T^{12} - 640 p^{12} T^{14} + p^{16} T^{16} \) |
| 17 | \( 1 - 1544 T^{2} + 1149112 T^{4} - 544758248 T^{6} + 183674027314 T^{8} - 544758248 p^{4} T^{10} + 1149112 p^{8} T^{12} - 1544 p^{12} T^{14} + p^{16} T^{16} \) |
| 19 | \( 1 - 1752 T^{2} + 1403532 T^{4} - 710782632 T^{6} + 278533058918 T^{8} - 710782632 p^{4} T^{10} + 1403532 p^{8} T^{12} - 1752 p^{12} T^{14} + p^{16} T^{16} \) |
| 23 | \( ( 1 - 8 T + 1504 T^{2} - 2840 T^{3} + 993562 T^{4} - 2840 p^{2} T^{5} + 1504 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 - 40 T + 1440 T^{2} - 35480 T^{3} + 1476410 T^{4} - 35480 p^{2} T^{5} + 1440 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 31 | \( 1 - 40 p T^{2} + 1929036 T^{4} - 80954072 p T^{6} + 1757674512806 T^{8} - 80954072 p^{5} T^{10} + 1929036 p^{8} T^{12} - 40 p^{13} T^{14} + p^{16} T^{16} \) |
| 37 | \( ( 1 - 64 T + 1984 T^{2} - 33664 T^{3} + 1137634 T^{4} - 33664 p^{2} T^{5} + 1984 p^{4} T^{6} - 64 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 41 | \( 1 - 3640 T^{2} + 14914392 T^{4} - 31945062104 T^{6} + 68419019701298 T^{8} - 31945062104 p^{4} T^{10} + 14914392 p^{8} T^{12} - 3640 p^{12} T^{14} + p^{16} T^{16} \) |
| 43 | \( ( 1 - 56 T + 84 p T^{2} - 107464 T^{3} + 6712358 T^{4} - 107464 p^{2} T^{5} + 84 p^{5} T^{6} - 56 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 47 | \( 1 - 6184 T^{2} + 19495500 T^{4} - 37677414104 T^{6} + 70771336814246 T^{8} - 37677414104 p^{4} T^{10} + 19495500 p^{8} T^{12} - 6184 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( ( 1 + 72 T + 5772 T^{2} + 118776 T^{3} + 8182022 T^{4} + 118776 p^{2} T^{5} + 5772 p^{4} T^{6} + 72 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 59 | \( 1 - 5160 T^{2} + 37946892 T^{4} - 155447065944 T^{6} + 676713989131238 T^{8} - 155447065944 p^{4} T^{10} + 37946892 p^{8} T^{12} - 5160 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( 1 - 16576 T^{2} + 105927552 T^{4} - 335063709632 T^{6} + 884820867334466 T^{8} - 335063709632 p^{4} T^{10} + 105927552 p^{8} T^{12} - 16576 p^{12} T^{14} + p^{16} T^{16} \) |
| 67 | \( ( 1 - 32 T + 11908 T^{2} - 166496 T^{3} + 66053926 T^{4} - 166496 p^{2} T^{5} + 11908 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 + 112 T + 13704 T^{2} + 1059392 T^{3} + 79617770 T^{4} + 1059392 p^{2} T^{5} + 13704 p^{4} T^{6} + 112 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 73 | \( 1 - 20160 T^{2} + 132997248 T^{4} - 74480341440 T^{6} - 2267376260578174 T^{8} - 74480341440 p^{4} T^{10} + 132997248 p^{8} T^{12} - 20160 p^{12} T^{14} + p^{16} T^{16} \) |
| 79 | \( ( 1 - 216 T + 37404 T^{2} - 4106472 T^{3} + 381926342 T^{4} - 4106472 p^{2} T^{5} + 37404 p^{4} T^{6} - 216 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 83 | \( 1 - 35400 T^{2} + 603816348 T^{4} - 6617387785848 T^{6} + 52566528325929350 T^{8} - 6617387785848 p^{4} T^{10} + 603816348 p^{8} T^{12} - 35400 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( 1 - 21768 T^{2} + 275422584 T^{4} - 2467902958824 T^{6} + 20047392847102130 T^{8} - 2467902958824 p^{4} T^{10} + 275422584 p^{8} T^{12} - 21768 p^{12} T^{14} + p^{16} T^{16} \) |
| 97 | \( 1 - 61632 T^{2} + 1744804416 T^{4} - 29860121715648 T^{6} + 339981119551810562 T^{8} - 29860121715648 p^{4} T^{10} + 1744804416 p^{8} T^{12} - 61632 p^{12} T^{14} + p^{16} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.46348192350281342010770507490, −3.21026163272829711463535935028, −3.17361903309831327944712097200, −3.11600890486832689125030173736, −3.09476466276875981259971393530, −2.95235339838814505610468722458, −2.71000278904675309850839466612, −2.68495994639364673023581082729, −2.66763804478774180690825065573, −2.64428582623676403016178881453, −2.36953952090554538756808384462, −2.31527058916099939561391320464, −2.08934052274153435671179810206, −1.77776435967649921461877405314, −1.76964229375938723403692476338, −1.54874242258754765082910830679, −1.44676785839624812346159548509, −1.19612211090973924904912158523, −1.06238817232626185232729214296, −0.846254784557119019019653920207, −0.792689322030821549052619484097, −0.72446675687005529927499621209, −0.53216630720172241954356599373, −0.52159861196825195571800730790, −0.22344843036424864980874629780,
0.22344843036424864980874629780, 0.52159861196825195571800730790, 0.53216630720172241954356599373, 0.72446675687005529927499621209, 0.792689322030821549052619484097, 0.846254784557119019019653920207, 1.06238817232626185232729214296, 1.19612211090973924904912158523, 1.44676785839624812346159548509, 1.54874242258754765082910830679, 1.76964229375938723403692476338, 1.77776435967649921461877405314, 2.08934052274153435671179810206, 2.31527058916099939561391320464, 2.36953952090554538756808384462, 2.64428582623676403016178881453, 2.66763804478774180690825065573, 2.68495994639364673023581082729, 2.71000278904675309850839466612, 2.95235339838814505610468722458, 3.09476466276875981259971393530, 3.11600890486832689125030173736, 3.17361903309831327944712097200, 3.21026163272829711463535935028, 3.46348192350281342010770507490
Plot not available for L-functions of degree greater than 10.