| L(s) = 1 | − 16·5-s − 12·9-s + 48·13-s + 16·17-s + 24·25-s − 80·29-s − 16·37-s + 80·41-s + 192·45-s + 152·49-s + 176·53-s − 272·61-s − 768·65-s − 16·73-s + 90·81-s − 256·85-s − 240·89-s + 400·97-s − 528·101-s + 560·109-s + 336·113-s − 576·117-s + 520·121-s + 1.04e3·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 3.19·5-s − 4/3·9-s + 3.69·13-s + 0.941·17-s + 0.959·25-s − 2.75·29-s − 0.432·37-s + 1.95·41-s + 4.26·45-s + 3.10·49-s + 3.32·53-s − 4.45·61-s − 11.8·65-s − 0.219·73-s + 10/9·81-s − 3.01·85-s − 2.69·89-s + 4.12·97-s − 5.22·101-s + 5.13·109-s + 2.97·113-s − 4.92·117-s + 4.29·121-s + 8.31·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4222018189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4222018189\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + p T^{2} )^{4} \) |
| good | 5 | \( ( 1 + 8 T + 84 T^{2} + 472 T^{3} + 2822 T^{4} + 472 p^{2} T^{5} + 84 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 7 | \( 1 - 152 T^{2} + 1476 p T^{4} - 514600 T^{6} + 25187654 T^{8} - 514600 p^{4} T^{10} + 1476 p^{9} T^{12} - 152 p^{12} T^{14} + p^{16} T^{16} \) |
| 11 | \( ( 1 - 260 T^{2} + 40038 T^{4} - 260 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - 24 T + 540 T^{2} - 5736 T^{3} + 89702 T^{4} - 5736 p^{2} T^{5} + 540 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 8 T + 252 T^{2} + 2888 T^{3} + 48902 T^{4} + 2888 p^{2} T^{5} + 252 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 19 | \( 1 - 1224 T^{2} + 1027804 T^{4} - 554589432 T^{6} + 236077316358 T^{8} - 554589432 p^{4} T^{10} + 1027804 p^{8} T^{12} - 1224 p^{12} T^{14} + p^{16} T^{16} \) |
| 23 | \( 1 - 3272 T^{2} + 5007132 T^{4} - 4715977336 T^{6} + 5669682278 p^{2} T^{8} - 4715977336 p^{4} T^{10} + 5007132 p^{8} T^{12} - 3272 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( ( 1 + 40 T + 2196 T^{2} + 31160 T^{3} + 1501766 T^{4} + 31160 p^{2} T^{5} + 2196 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 31 | \( 1 - 4248 T^{2} + 9652444 T^{4} - 14917063464 T^{6} + 16677690731718 T^{8} - 14917063464 p^{4} T^{10} + 9652444 p^{8} T^{12} - 4248 p^{12} T^{14} + p^{16} T^{16} \) |
| 37 | \( ( 1 + 8 T + 1980 T^{2} - 54920 T^{3} + 1119206 T^{4} - 54920 p^{2} T^{5} + 1980 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 - 40 T + 2556 T^{2} + 43240 T^{3} - 289594 T^{4} + 43240 p^{2} T^{5} + 2556 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 43 | \( 1 - 5960 T^{2} + 23041116 T^{4} - 59814611320 T^{6} + 126481252740230 T^{8} - 59814611320 p^{4} T^{10} + 23041116 p^{8} T^{12} - 5960 p^{12} T^{14} + p^{16} T^{16} \) |
| 47 | \( 1 - 12360 T^{2} + 70380700 T^{4} - 250403346936 T^{6} + 638353123484742 T^{8} - 250403346936 p^{4} T^{10} + 70380700 p^{8} T^{12} - 12360 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( ( 1 - 88 T + 10068 T^{2} - 520520 T^{3} + 37632134 T^{4} - 520520 p^{2} T^{5} + 10068 p^{4} T^{6} - 88 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 59 | \( 1 - 7688 T^{2} + 37931484 T^{4} - 124184601784 T^{6} + 382130398823558 T^{8} - 124184601784 p^{4} T^{10} + 37931484 p^{8} T^{12} - 7688 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 + 136 T + 14076 T^{2} + 1001336 T^{3} + 63819302 T^{4} + 1001336 p^{2} T^{5} + 14076 p^{4} T^{6} + 136 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 - 8644 T^{2} + 58097190 T^{4} - 8644 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( 1 - 18120 T^{2} + 152088604 T^{4} - 903092412024 T^{6} + 4737261672397254 T^{8} - 903092412024 p^{4} T^{10} + 152088604 p^{8} T^{12} - 18120 p^{12} T^{14} + p^{16} T^{16} \) |
| 73 | \( ( 1 + 8 T + 5212 T^{2} + 653240 T^{3} - 3523322 T^{4} + 653240 p^{2} T^{5} + 5212 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 79 | \( 1 - 26776 T^{2} + 333200860 T^{4} - 2815930225960 T^{6} + 19175645821896646 T^{8} - 2815930225960 p^{4} T^{10} + 333200860 p^{8} T^{12} - 26776 p^{12} T^{14} + p^{16} T^{16} \) |
| 83 | \( 1 - 15752 T^{2} + 70662492 T^{4} - 102795674680 T^{6} + 590797586843654 T^{8} - 102795674680 p^{4} T^{10} + 70662492 p^{8} T^{12} - 15752 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( ( 1 + 120 T + 30300 T^{2} + 2699976 T^{3} + 352873862 T^{4} + 2699976 p^{2} T^{5} + 30300 p^{4} T^{6} + 120 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 200 T + 21276 T^{2} - 3009400 T^{3} + 385130822 T^{4} - 3009400 p^{2} T^{5} + 21276 p^{4} T^{6} - 200 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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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
−4.75614335748711971211001773386, −4.61382630628657586024668580620, −4.37972897683909811264559094139, −4.11570379484068911403684601267, −4.09937569932233319517812426338, −3.99476278487736800343135389777, −3.83150006022524197812799067257, −3.74253956451734473236375963582, −3.66026410980715678240478978575, −3.46537087984794372110121460785, −3.33467711206721674766703980120, −3.28924334526662370943616769679, −3.00992939595715092188120593113, −2.98617778780051879919517157990, −2.50223000205599719489343386914, −2.26585267980616344166614683539, −2.13923115926971451085715332291, −2.04243914786588183805626238513, −1.82764263738140355806353544754, −1.38939228300423989026835517459, −1.12211401472594293858186585503, −0.972446993435436141898898757880, −0.813620279231900163503385192624, −0.27658940651158177774747829322, −0.15882316132642915296889295001,
0.15882316132642915296889295001, 0.27658940651158177774747829322, 0.813620279231900163503385192624, 0.972446993435436141898898757880, 1.12211401472594293858186585503, 1.38939228300423989026835517459, 1.82764263738140355806353544754, 2.04243914786588183805626238513, 2.13923115926971451085715332291, 2.26585267980616344166614683539, 2.50223000205599719489343386914, 2.98617778780051879919517157990, 3.00992939595715092188120593113, 3.28924334526662370943616769679, 3.33467711206721674766703980120, 3.46537087984794372110121460785, 3.66026410980715678240478978575, 3.74253956451734473236375963582, 3.83150006022524197812799067257, 3.99476278487736800343135389777, 4.09937569932233319517812426338, 4.11570379484068911403684601267, 4.37972897683909811264559094139, 4.61382630628657586024668580620, 4.75614335748711971211001773386
Plot not available for L-functions of degree greater than 10.
