| L(s) = 1 | − 32·11-s + 80·17-s − 56·19-s + 92·25-s − 128·41-s − 28·49-s + 104·59-s − 304·67-s − 112·73-s + 72·83-s + 512·89-s + 64·97-s + 688·107-s − 136·113-s + 112·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 444·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 2.90·11-s + 4.70·17-s − 2.94·19-s + 3.67·25-s − 3.12·41-s − 4/7·49-s + 1.76·59-s − 4.53·67-s − 1.53·73-s + 0.867·83-s + 5.75·89-s + 0.659·97-s + 6.42·107-s − 1.20·113-s + 0.925·121-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 + 2.62·169-s + 0.00578·173-s + 0.00558·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(13.81430227\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.81430227\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 + p T^{2} )^{4} \) |
| good | 5 | \( 1 - 92 T^{2} + 5464 T^{4} - 211956 T^{6} + 6231214 T^{8} - 211956 p^{4} T^{10} + 5464 p^{8} T^{12} - 92 p^{12} T^{14} + p^{16} T^{16} \) |
| 11 | \( ( 1 + 16 T + 328 T^{2} + 4944 T^{3} + 49230 T^{4} + 4944 p^{2} T^{5} + 328 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 13 | \( 1 - 444 T^{2} + 142936 T^{4} - 35361044 T^{6} + 6575436334 T^{8} - 35361044 p^{4} T^{10} + 142936 p^{8} T^{12} - 444 p^{12} T^{14} + p^{16} T^{16} \) |
| 17 | \( ( 1 - 40 T + 1308 T^{2} - 31512 T^{3} + 588230 T^{4} - 31512 p^{2} T^{5} + 1308 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 + 28 T + 90 p T^{2} + 31332 T^{3} + 975266 T^{4} + 31332 p^{2} T^{5} + 90 p^{5} T^{6} + 28 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 23 | \( 1 - 1744 T^{2} + 1272156 T^{4} - 412076080 T^{6} + 99307893702 T^{8} - 412076080 p^{4} T^{10} + 1272156 p^{8} T^{12} - 1744 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( 1 - 3384 T^{2} + 6555580 T^{4} - 8754768776 T^{6} + 8490907402822 T^{8} - 8754768776 p^{4} T^{10} + 6555580 p^{8} T^{12} - 3384 p^{12} T^{14} + p^{16} T^{16} \) |
| 31 | \( 1 - 3944 T^{2} + 8438620 T^{4} - 12447428312 T^{6} + 13694235978694 T^{8} - 12447428312 p^{4} T^{10} + 8438620 p^{8} T^{12} - 3944 p^{12} T^{14} + p^{16} T^{16} \) |
| 37 | \( 1 - 3512 T^{2} + 9188668 T^{4} - 18622781448 T^{6} + 27544347275206 T^{8} - 18622781448 p^{4} T^{10} + 9188668 p^{8} T^{12} - 3512 p^{12} T^{14} + p^{16} T^{16} \) |
| 41 | \( ( 1 + 64 T + 4956 T^{2} + 221760 T^{3} + 11848326 T^{4} + 221760 p^{2} T^{5} + 4956 p^{4} T^{6} + 64 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 + 4680 T^{2} + 58016 T^{3} + 10251086 T^{4} + 58016 p^{2} T^{5} + 4680 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( 1 - 8392 T^{2} + 39566748 T^{4} - 127207295352 T^{6} + 316693927920198 T^{8} - 127207295352 p^{4} T^{10} + 39566748 p^{8} T^{12} - 8392 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( 1 - 18920 T^{2} + 162796828 T^{4} - 840091728600 T^{6} + 2864724835962118 T^{8} - 840091728600 p^{4} T^{10} + 162796828 p^{8} T^{12} - 18920 p^{12} T^{14} + p^{16} T^{16} \) |
| 59 | \( ( 1 - 52 T + 2254 T^{2} + 207508 T^{3} - 19795230 T^{4} + 207508 p^{2} T^{5} + 2254 p^{4} T^{6} - 52 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 61 | \( 1 - 16316 T^{2} + 140172120 T^{4} - 816942037524 T^{6} + 3499102878259502 T^{8} - 816942037524 p^{4} T^{10} + 140172120 p^{8} T^{12} - 16316 p^{12} T^{14} + p^{16} T^{16} \) |
| 67 | \( ( 1 + 152 T + 22224 T^{2} + 2037320 T^{3} + 158433022 T^{4} + 2037320 p^{2} T^{5} + 22224 p^{4} T^{6} + 152 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 71 | \( 1 - 9864 T^{2} + 51888284 T^{4} - 294531431096 T^{6} + 1789421441990854 T^{8} - 294531431096 p^{4} T^{10} + 51888284 p^{8} T^{12} - 9864 p^{12} T^{14} + p^{16} T^{16} \) |
| 73 | \( ( 1 + 56 T + 18460 T^{2} + 736008 T^{3} + 138223494 T^{4} + 736008 p^{2} T^{5} + 18460 p^{4} T^{6} + 56 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 79 | \( 1 - 24968 T^{2} + 330869788 T^{4} - 3106127956152 T^{6} + 22189846569597766 T^{8} - 3106127956152 p^{4} T^{10} + 330869788 p^{8} T^{12} - 24968 p^{12} T^{14} + p^{16} T^{16} \) |
| 83 | \( ( 1 - 36 T + 16478 T^{2} - 177884 T^{3} + 135298114 T^{4} - 177884 p^{2} T^{5} + 16478 p^{4} T^{6} - 36 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 - 256 T + 48252 T^{2} - 6269952 T^{3} + 638304966 T^{4} - 6269952 p^{2} T^{5} + 48252 p^{4} T^{6} - 256 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 32 T + 19484 T^{2} - 1437536 T^{3} + 199130566 T^{4} - 1437536 p^{2} T^{5} + 19484 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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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
−3.40909335537205051798306060106, −3.40808190221764914847625008218, −3.37600615895400312589535923754, −3.33143768331963988976705474720, −3.09512576248141533557783234193, −3.04719623536023208327151413431, −2.93489128069278887996081964948, −2.80473649165457589576383275832, −2.76411357664872317780444643687, −2.48923456702483553819259175535, −2.28599692458137821799765005569, −2.23822600435032895747215210825, −2.06923985667416994283376423661, −2.03801334447952913018176326746, −1.86752616822704202667864214092, −1.62791304818724949077049542910, −1.41800380540568796716613007174, −1.27896221416979120014107971004, −1.23320069341065856239590454522, −1.06994358405321471640389013242, −0.819211676704466744129224403192, −0.49132977639356202104552956938, −0.45180366703321534338482215485, −0.40462907144439652535781800528, −0.26795131996062980569972746771,
0.26795131996062980569972746771, 0.40462907144439652535781800528, 0.45180366703321534338482215485, 0.49132977639356202104552956938, 0.819211676704466744129224403192, 1.06994358405321471640389013242, 1.23320069341065856239590454522, 1.27896221416979120014107971004, 1.41800380540568796716613007174, 1.62791304818724949077049542910, 1.86752616822704202667864214092, 2.03801334447952913018176326746, 2.06923985667416994283376423661, 2.23822600435032895747215210825, 2.28599692458137821799765005569, 2.48923456702483553819259175535, 2.76411357664872317780444643687, 2.80473649165457589576383275832, 2.93489128069278887996081964948, 3.04719623536023208327151413431, 3.09512576248141533557783234193, 3.33143768331963988976705474720, 3.37600615895400312589535923754, 3.40808190221764914847625008218, 3.40909335537205051798306060106
Plot not available for L-functions of degree greater than 10.
