| L(s) = 1 | − 2·9-s − 22·25-s + 4·37-s − 20·43-s + 28·67-s + 56·79-s − 13·81-s + 92·109-s + 98·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 98·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 2/3·9-s − 4.39·25-s + 0.657·37-s − 3.04·43-s + 3.42·67-s + 6.30·79-s − 1.44·81-s + 8.81·109-s + 8.90·121-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 + 7.53·169-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 + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(20.48197589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(20.48197589\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 2 T^{2} + 17 T^{4} + 46 T^{6} + 172 T^{8} + 46 p^{2} T^{10} + 17 p^{4} T^{12} + 2 p^{6} T^{14} + p^{8} T^{16} \) |
| 7 | \( 1 \) |
| good | 5 | \( ( 1 + 11 T^{2} + 93 T^{4} + 622 T^{6} + 3086 T^{8} + 622 p^{2} T^{10} + 93 p^{4} T^{12} + 11 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 11 | \( ( 1 - 49 T^{2} + 1101 T^{4} - 15842 T^{6} + 183302 T^{8} - 15842 p^{2} T^{10} + 1101 p^{4} T^{12} - 49 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 13 | \( ( 1 - 49 T^{2} + 1278 T^{4} - 23495 T^{6} + 341186 T^{8} - 23495 p^{2} T^{10} + 1278 p^{4} T^{12} - 49 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 17 | \( ( 1 + 42 T^{2} + 705 T^{4} + 11358 T^{6} + 224396 T^{8} + 11358 p^{2} T^{10} + 705 p^{4} T^{12} + 42 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 19 | \( ( 1 - 115 T^{2} + 6345 T^{4} - 216074 T^{6} + 4950494 T^{8} - 216074 p^{2} T^{10} + 6345 p^{4} T^{12} - 115 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 23 | \( ( 1 - 102 T^{2} + 4081 T^{4} - 83682 T^{6} + 1454124 T^{8} - 83682 p^{2} T^{10} + 4081 p^{4} T^{12} - 102 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 29 | \( ( 1 - 103 T^{2} + 6606 T^{4} - 293969 T^{6} + 9810626 T^{8} - 293969 p^{2} T^{10} + 6606 p^{4} T^{12} - 103 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 31 | \( ( 1 - 4 p T^{2} + 8778 T^{4} - 418928 T^{6} + 15011795 T^{8} - 418928 p^{2} T^{10} + 8778 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} )^{2} \) |
| 37 | \( ( 1 - T + 61 T^{2} + 314 T^{3} + 1294 T^{4} + 314 p T^{5} + 61 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 41 | \( ( 1 + 240 T^{2} + 27996 T^{4} + 2035728 T^{6} + 100303238 T^{8} + 2035728 p^{2} T^{10} + 27996 p^{4} T^{12} + 240 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 43 | \( ( 1 + 5 T + 76 T^{2} + 341 T^{3} + 2710 T^{4} + 341 p T^{5} + 76 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 47 | \( ( 1 + 158 T^{2} + 11793 T^{4} + 733630 T^{6} + 39908708 T^{8} + 733630 p^{2} T^{10} + 11793 p^{4} T^{12} + 158 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 53 | \( ( 1 - 5 p T^{2} + 37257 T^{4} - 3352538 T^{6} + 211268186 T^{8} - 3352538 p^{2} T^{10} + 37257 p^{4} T^{12} - 5 p^{7} T^{14} + p^{8} T^{16} )^{2} \) |
| 59 | \( ( 1 + 187 T^{2} + 23065 T^{4} + 2022838 T^{6} + 2291606 p T^{8} + 2022838 p^{2} T^{10} + 23065 p^{4} T^{12} + 187 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 61 | \( ( 1 - 338 T^{2} + 55633 T^{4} - 5801762 T^{6} + 419624596 T^{8} - 5801762 p^{2} T^{10} + 55633 p^{4} T^{12} - 338 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 67 | \( ( 1 - 7 T + 249 T^{2} - 1304 T^{3} + 24614 T^{4} - 1304 p T^{5} + 249 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 71 | \( ( 1 - 224 T^{2} + 21244 T^{4} - 988448 T^{6} + 37865158 T^{8} - 988448 p^{2} T^{10} + 21244 p^{4} T^{12} - 224 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 73 | \( ( 1 - 399 T^{2} + 70689 T^{4} - 7728594 T^{6} + 628610462 T^{8} - 7728594 p^{2} T^{10} + 70689 p^{4} T^{12} - 399 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 79 | \( ( 1 - 14 T + 348 T^{2} - 40 p T^{3} + 42101 T^{4} - 40 p^{2} T^{5} + 348 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 83 | \( ( 1 + 141 T^{2} + 22278 T^{4} + 1798779 T^{6} + 189218258 T^{8} + 1798779 p^{2} T^{10} + 22278 p^{4} T^{12} + 141 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 89 | \( ( 1 + 378 T^{2} + 73089 T^{4} + 9573054 T^{6} + 954583820 T^{8} + 9573054 p^{2} T^{10} + 73089 p^{4} T^{12} + 378 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 97 | \( ( 1 - 429 T^{2} + 87258 T^{4} - 11450019 T^{6} + 1191212138 T^{8} - 11450019 p^{2} T^{10} + 87258 p^{4} T^{12} - 429 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
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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
−2.15727162772985858481669936498, −2.13868133204960656589432737325, −2.02559777468044963754252426312, −1.95992545846675926763256217062, −1.90770214229428223939693444810, −1.88115341868601045010403604718, −1.81539351063710961057977383345, −1.76546428110081210222395261546, −1.74152896098541145829522396347, −1.68616452846801796642719937008, −1.60468629031090174308396807701, −1.49287493733244921066351227714, −1.42816561843174125948019718418, −1.24717333038836533399089220726, −0.957288950607571373136951945901, −0.920768721489632762991324679223, −0.882079108738514981639563985546, −0.73609134197326094903676878877, −0.63373484043106877796165691784, −0.62123307177165014629476828677, −0.61023505628231267208804643012, −0.55355520683835593838245262814, −0.45650115994225670298849661067, −0.36932741822865904250285318216, −0.10655647260328114732855199197,
0.10655647260328114732855199197, 0.36932741822865904250285318216, 0.45650115994225670298849661067, 0.55355520683835593838245262814, 0.61023505628231267208804643012, 0.62123307177165014629476828677, 0.63373484043106877796165691784, 0.73609134197326094903676878877, 0.882079108738514981639563985546, 0.920768721489632762991324679223, 0.957288950607571373136951945901, 1.24717333038836533399089220726, 1.42816561843174125948019718418, 1.49287493733244921066351227714, 1.60468629031090174308396807701, 1.68616452846801796642719937008, 1.74152896098541145829522396347, 1.76546428110081210222395261546, 1.81539351063710961057977383345, 1.88115341868601045010403604718, 1.90770214229428223939693444810, 1.95992545846675926763256217062, 2.02559777468044963754252426312, 2.13868133204960656589432737325, 2.15727162772985858481669936498
Plot not available for L-functions of degree greater than 10.
