| L(s) = 1 | + 12·4-s − 4·9-s + 88·16-s + 48·19-s + 344·31-s − 48·36-s − 104·49-s − 392·61-s + 432·64-s + 576·76-s + 608·79-s − 74·81-s + 208·109-s − 668·121-s + 4.12e3·124-s + 127-s + 131-s + 137-s + 139-s − 352·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 752·169-s − 192·171-s + ⋯ |
| L(s) = 1 | + 3·4-s − 4/9·9-s + 11/2·16-s + 2.52·19-s + 11.0·31-s − 4/3·36-s − 2.12·49-s − 6.42·61-s + 27/4·64-s + 7.57·76-s + 7.69·79-s − 0.913·81-s + 1.90·109-s − 5.52·121-s + 33.2·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 2.44·144-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 4.44·169-s − 1.12·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(32.38714423\) |
| \(L(\frac12)\) |
\(\approx\) |
\(32.38714423\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 4 T^{2} + 10 p^{2} T^{4} - 944 T^{6} - 2381 T^{8} - 944 p^{4} T^{10} + 10 p^{10} T^{12} + 4 p^{12} T^{14} + p^{16} T^{16} \) |
| 5 | \( 1 + 2 p^{3} T^{4} - 21 p^{6} T^{8} + 2 p^{11} T^{12} + p^{16} T^{16} \) |
| 7 | \( ( 1 + 52 T^{2} + 87 p^{2} T^{4} + 52 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| good | 2 | \( ( 1 - 3 p T^{2} + 5 p T^{4} + 9 p^{2} T^{6} - 39 p^{2} T^{8} + 9 p^{6} T^{10} + 5 p^{9} T^{12} - 3 p^{13} T^{14} + p^{16} T^{16} )^{2} \) |
| 11 | \( ( 1 + 334 T^{2} + 57760 T^{4} + 8187676 T^{6} + 1046359339 T^{8} + 8187676 p^{4} T^{10} + 57760 p^{8} T^{12} + 334 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 13 | \( ( 1 - 188 T^{2} + 20583 T^{4} - 188 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 17 | \( ( 1 - 906 T^{2} + 463000 T^{4} - 172859364 T^{6} + 52513801899 T^{8} - 172859364 p^{4} T^{10} + 463000 p^{8} T^{12} - 906 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 19 | \( ( 1 - 12 T - 599 T^{2} - 252 T^{3} + 369744 T^{4} - 252 p^{2} T^{5} - 599 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 23 | \( ( 1 - 1066 T^{2} + 315400 T^{4} - 278518084 T^{6} + 277658769259 T^{8} - 278518084 p^{4} T^{10} + 315400 p^{8} T^{12} - 1066 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 29 | \( ( 1 - 2394 T^{2} + 2753756 T^{4} - 2394 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 31 | \( ( 1 - 86 T + 3685 T^{2} - 153854 T^{3} + 5740444 T^{4} - 153854 p^{2} T^{5} + 3685 p^{4} T^{6} - 86 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 37 | \( ( 1 + 2468 T^{2} + 1780081 T^{4} + 1388548628 T^{6} + 3656190464464 T^{8} + 1388548628 p^{4} T^{10} + 1780081 p^{8} T^{12} + 2468 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 41 | \( ( 1 - 3054 T^{2} + 6815636 T^{4} - 3054 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 43 | \( ( 1 - 4124 T^{2} + 11073111 T^{4} - 4124 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 47 | \( ( 1 - 7336 T^{2} + 31122250 T^{4} - 94893243424 T^{6} + 228626385513379 T^{8} - 94893243424 p^{4} T^{10} + 31122250 p^{8} T^{12} - 7336 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 53 | \( ( 1 - 7636 T^{2} + 28650250 T^{4} - 105966940624 T^{6} + 357269785558579 T^{8} - 105966940624 p^{4} T^{10} + 28650250 p^{8} T^{12} - 7636 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 59 | \( ( 1 + 474 T^{2} + 12860200 T^{4} - 17476496604 T^{6} + 10103548950219 T^{8} - 17476496604 p^{4} T^{10} + 12860200 p^{8} T^{12} + 474 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 61 | \( ( 1 + 98 T + 136 T^{2} + 198548 T^{3} + 40060699 T^{4} + 198548 p^{2} T^{5} + 136 p^{4} T^{6} + 98 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 67 | \( ( 1 + 12548 T^{2} + 81642721 T^{4} + 445546114868 T^{6} + 2168184767444464 T^{8} + 445546114868 p^{4} T^{10} + 81642721 p^{8} T^{12} + 12548 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 71 | \( ( 1 - 13674 T^{2} + 87943916 T^{4} - 13674 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 73 | \( ( 1 + 10828 T^{2} + 58596841 T^{4} + 20056282108 T^{6} - 696353387031776 T^{8} + 20056282108 p^{4} T^{10} + 58596841 p^{8} T^{12} + 10828 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 79 | \( ( 1 - 152 T + 4981 T^{2} - 857432 T^{3} + 145300984 T^{4} - 857432 p^{2} T^{5} + 4981 p^{4} T^{6} - 152 p^{6} T^{7} + p^{8} T^{8} )^{4} \) |
| 83 | \( ( 1 + 7386 T^{2} + 83632076 T^{4} + 7386 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| 89 | \( ( 1 - 1586 T^{2} - 121555520 T^{4} + 2241915676 T^{6} + 11299181809334539 T^{8} + 2241915676 p^{4} T^{10} - 121555520 p^{8} T^{12} - 1586 p^{12} T^{14} + p^{16} T^{16} )^{2} \) |
| 97 | \( ( 1 - 37448 T^{2} + 527638098 T^{4} - 37448 p^{4} T^{6} + p^{8} T^{8} )^{4} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.89582835566798559926186319547, −3.71714283654786154020038916265, −3.56160032952795522618213981749, −3.47693574817576494567709102314, −3.39343065907656789796931377177, −3.33034808235857770858893882388, −3.15789989213865363061611904673, −3.06567165373098373285659821319, −2.93106739520428817967027018980, −2.81785623249694407595296026734, −2.79490619265118629517914206702, −2.77631226923327885045307993238, −2.60233703092335750155863565304, −2.42411786118666289230701219657, −2.37525987648620775742984734512, −2.15285452680164303372314334841, −2.01127520909562254602281258748, −1.96420466559638108415540453247, −1.41629332604352798620029223162, −1.31039483097622593488844790063, −1.29010157416730295557730040311, −1.18291282778892749501470561703, −1.07440162696704758806588195137, −0.73295629672955769027688904652, −0.43035754040625187678929390224,
0.43035754040625187678929390224, 0.73295629672955769027688904652, 1.07440162696704758806588195137, 1.18291282778892749501470561703, 1.29010157416730295557730040311, 1.31039483097622593488844790063, 1.41629332604352798620029223162, 1.96420466559638108415540453247, 2.01127520909562254602281258748, 2.15285452680164303372314334841, 2.37525987648620775742984734512, 2.42411786118666289230701219657, 2.60233703092335750155863565304, 2.77631226923327885045307993238, 2.79490619265118629517914206702, 2.81785623249694407595296026734, 2.93106739520428817967027018980, 3.06567165373098373285659821319, 3.15789989213865363061611904673, 3.33034808235857770858893882388, 3.39343065907656789796931377177, 3.47693574817576494567709102314, 3.56160032952795522618213981749, 3.71714283654786154020038916265, 3.89582835566798559926186319547
Plot not available for L-functions of degree greater than 10.
