L(s) = 1 | − 6·4-s + 17·16-s + 2·25-s + 24·29-s − 20·49-s + 8·61-s − 40·64-s − 64·79-s − 12·100-s − 8·101-s − 144·116-s + 92·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
L(s) = 1 | − 3·4-s + 17/4·16-s + 2/5·25-s + 4.45·29-s − 2.85·49-s + 1.02·61-s − 5·64-s − 7.20·79-s − 6/5·100-s − 0.796·101-s − 13.3·116-s + 8.36·121-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 − 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.122906498\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.122906498\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - 2 T^{2} + 27 T^{4} - 164 T^{6} + 27 p^{2} T^{8} - 2 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( 1 + 2 p T^{2} + 343 T^{4} + 3500 T^{6} + 343 p^{2} T^{8} + 2 p^{5} T^{10} + p^{6} T^{12} \) |
good | 2 | \( ( 1 + 3 T^{2} + 5 T^{4} + 11 T^{6} + 5 p^{2} T^{8} + 3 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 7 | \( ( 1 + 10 T^{2} + 47 T^{4} + 108 T^{6} + 47 p^{2} T^{8} + 10 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 11 | \( ( 1 - 46 T^{2} + 1019 T^{4} - 13884 T^{6} + 1019 p^{2} T^{8} - 46 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 17 | \( ( 1 - 14 T^{2} + 543 T^{4} - 3812 T^{6} + 543 p^{2} T^{8} - 14 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 19 | \( ( 1 - 50 T^{2} + 1783 T^{4} - 38972 T^{6} + 1783 p^{2} T^{8} - 50 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 23 | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{6} \) |
| 29 | \( ( 1 - 6 T + 59 T^{2} - 244 T^{3} + 59 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 31 | \( ( 1 - 26 T^{2} + 799 T^{4} - 15884 T^{6} + 799 p^{2} T^{8} - 26 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 37 | \( ( 1 + 90 T^{2} + 3303 T^{4} + 99052 T^{6} + 3303 p^{2} T^{8} + 90 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 41 | \( ( 1 - 170 T^{2} + 14211 T^{4} - 730580 T^{6} + 14211 p^{2} T^{8} - 170 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 43 | \( ( 1 - 186 T^{2} + 16007 T^{4} - 847532 T^{6} + 16007 p^{2} T^{8} - 186 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 47 | \( ( 1 + 262 T^{2} + 29459 T^{4} + 1819212 T^{6} + 29459 p^{2} T^{8} + 262 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 53 | \( ( 1 - 214 T^{2} + 23303 T^{4} - 1540020 T^{6} + 23303 p^{2} T^{8} - 214 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 59 | \( ( 1 - 190 T^{2} + 22011 T^{4} - 1543900 T^{6} + 22011 p^{2} T^{8} - 190 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 61 | \( ( 1 - 2 T + 87 T^{2} - 500 T^{3} + 87 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 67 | \( ( 1 + 162 T^{2} + 13079 T^{4} + 898652 T^{6} + 13079 p^{2} T^{8} + 162 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 71 | \( ( 1 + 42 T^{2} + 13187 T^{4} + 344404 T^{6} + 13187 p^{2} T^{8} + 42 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 73 | \( ( 1 + 386 T^{2} + 65263 T^{4} + 6191420 T^{6} + 65263 p^{2} T^{8} + 386 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 79 | \( ( 1 + 16 T + 289 T^{2} + 2560 T^{3} + 289 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 83 | \( ( 1 + 270 T^{2} + 37067 T^{4} + 3564092 T^{6} + 37067 p^{2} T^{8} + 270 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 89 | \( ( 1 - 394 T^{2} + 74019 T^{4} - 8327188 T^{6} + 74019 p^{2} T^{8} - 394 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 97 | \( ( 1 + 242 T^{2} + 40447 T^{4} + 4478108 T^{6} + 40447 p^{2} T^{8} + 242 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.39510158837317672688909278541, −3.32886793342450437273359967647, −3.28452211414885177149347166338, −3.26980582096269635555467852833, −3.10609098678954443671238728962, −3.07263047152314459736234566182, −2.95156186394459922381485905124, −2.90609945819445959736187370985, −2.72532535148245931650823414259, −2.63567476649855579623684330618, −2.43779976660373238648622911108, −2.36787998191170681140052443864, −2.32026958247137184680598651489, −2.07057693874660876198648324287, −1.93340766946558676823254744723, −1.67286912125523665239673530242, −1.54556426665050916525040912762, −1.41517636774632104427424108214, −1.40867767163576142209145769071, −1.30437283698218140755700925701, −1.06526993295224760300199083435, −0.72227736739406260464617562700, −0.57550655231728088314723537842, −0.48156297925504725042139747796, −0.20464206153874141013601962457,
0.20464206153874141013601962457, 0.48156297925504725042139747796, 0.57550655231728088314723537842, 0.72227736739406260464617562700, 1.06526993295224760300199083435, 1.30437283698218140755700925701, 1.40867767163576142209145769071, 1.41517636774632104427424108214, 1.54556426665050916525040912762, 1.67286912125523665239673530242, 1.93340766946558676823254744723, 2.07057693874660876198648324287, 2.32026958247137184680598651489, 2.36787998191170681140052443864, 2.43779976660373238648622911108, 2.63567476649855579623684330618, 2.72532535148245931650823414259, 2.90609945819445959736187370985, 2.95156186394459922381485905124, 3.07263047152314459736234566182, 3.10609098678954443671238728962, 3.26980582096269635555467852833, 3.28452211414885177149347166338, 3.32886793342450437273359967647, 3.39510158837317672688909278541
Plot not available for L-functions of degree greater than 10.