| L(s) = 1 | − 4·7-s + 8·13-s − 8·16-s + 20·19-s + 44·31-s − 28·37-s − 12·43-s + 2·49-s − 32·61-s − 40·67-s + 44·73-s + 16·79-s − 32·91-s − 52·97-s − 28·109-s + 32·112-s − 36·121-s + 127-s + 131-s − 80·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 2.21·13-s − 2·16-s + 4.58·19-s + 7.90·31-s − 4.60·37-s − 1.82·43-s + 2/7·49-s − 4.09·61-s − 4.88·67-s + 5.14·73-s + 1.80·79-s − 3.35·91-s − 5.27·97-s − 2.68·109-s + 3.02·112-s − 3.27·121-s + 0.0887·127-s + 0.0873·131-s − 6.93·133-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.099901790\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.099901790\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( ( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| good | 2 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2}( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | \( 1 - 4 T^{4} - 474 T^{8} - 4 p^{4} T^{12} + p^{8} T^{16} \) |
| 7 | \( ( 1 + 2 T + 5 T^{2} + 18 T^{3} + 8 T^{4} + 18 p T^{5} + 5 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 + 36 T^{2} + 746 T^{4} + 11304 T^{6} + 136227 T^{8} + 11304 p^{2} T^{10} + 746 p^{4} T^{12} + 36 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( 1 - 20 T^{2} + 154 T^{4} + 6640 T^{6} - 138605 T^{8} + 6640 p^{2} T^{10} + 154 p^{4} T^{12} - 20 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 - 10 T + 74 T^{2} - 384 T^{3} + 1871 T^{4} - 384 p T^{5} + 74 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 40 T^{2} + 1071 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 56 T^{2} + 2295 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 11 T + p T^{2} )^{4}( 1 + 59 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | \( ( 1 + 14 T + 50 T^{2} - 540 T^{3} - 6169 T^{4} - 540 p T^{5} + 50 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 - 48 T^{2} - 34 T^{4} + 38496 T^{6} - 891549 T^{8} + 38496 p^{2} T^{10} - 34 p^{4} T^{12} - 48 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 + 6 T + 65 T^{2} + 318 T^{3} + 1476 T^{4} + 318 p T^{5} + 65 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 + 476 T^{4} + 4588806 T^{8} + 476 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 100 T^{2} + 7686 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 - 180 T^{2} + 16106 T^{4} - 955080 T^{6} + 53631075 T^{8} - 955080 p^{2} T^{10} + 16106 p^{4} T^{12} - 180 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 + 16 T + 97 T^{2} + 592 T^{3} + 5944 T^{4} + 592 p T^{5} + 97 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 20 T + 221 T^{2} + 1740 T^{3} + 13436 T^{4} + 1740 p T^{5} + 221 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( 1 - 420 T^{2} + 83258 T^{4} - 10272360 T^{6} + 869192883 T^{8} - 10272360 p^{2} T^{10} + 83258 p^{4} T^{12} - 420 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( ( 1 - 22 T + 242 T^{2} - 2640 T^{3} + 26591 T^{4} - 2640 p T^{5} + 242 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 4 T + 135 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( 1 + 12068 T^{4} + 72822630 T^{8} + 12068 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( 1 + 192 T^{2} + 20798 T^{4} + 1633920 T^{6} + 107011107 T^{8} + 1633920 p^{2} T^{10} + 20798 p^{4} T^{12} + 192 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( ( 1 + 26 T + 233 T^{2} + 6 T^{3} - 14224 T^{4} + 6 p T^{5} + 233 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−6.26823562621844535330467014374, −6.26495207537446421593941058629, −6.20000484249232772546673685194, −6.11435758023908675058791089821, −5.44543275254098123184577273261, −5.39696761348663395393573123108, −5.38849914120836813171592587078, −5.36276844297827598036891700080, −4.84011772685990671517079182167, −4.83591734425286094627685709194, −4.60178722560852836747103706837, −4.42409989773184727371241538589, −4.33220781145468761107111636161, −3.97228162767179573397743756002, −3.71693415295866371271252561843, −3.45270564034734209378757651124, −3.26183312459923570166888887902, −3.03532220408995346974800512811, −3.00192904825385607603993818058, −2.93243566888240504604187183158, −2.69233883171238516764707234171, −2.09688643993328007616577316613, −1.49998061370157750183068441136, −1.42332209345856480948375067618, −0.985285673803616486930511727499,
0.985285673803616486930511727499, 1.42332209345856480948375067618, 1.49998061370157750183068441136, 2.09688643993328007616577316613, 2.69233883171238516764707234171, 2.93243566888240504604187183158, 3.00192904825385607603993818058, 3.03532220408995346974800512811, 3.26183312459923570166888887902, 3.45270564034734209378757651124, 3.71693415295866371271252561843, 3.97228162767179573397743756002, 4.33220781145468761107111636161, 4.42409989773184727371241538589, 4.60178722560852836747103706837, 4.83591734425286094627685709194, 4.84011772685990671517079182167, 5.36276844297827598036891700080, 5.38849914120836813171592587078, 5.39696761348663395393573123108, 5.44543275254098123184577273261, 6.11435758023908675058791089821, 6.20000484249232772546673685194, 6.26495207537446421593941058629, 6.26823562621844535330467014374
Plot not available for L-functions of degree greater than 10.
