| L(s) = 1 | − 4·3-s + 4-s + 6·9-s − 4·12-s + 4·16-s + 2·17-s + 8·23-s + 14·25-s − 2·29-s + 6·36-s + 10·43-s − 16·48-s − 15·49-s − 8·51-s + 44·53-s − 32·61-s + 3·64-s + 2·68-s − 32·69-s − 56·75-s + 60·79-s − 15·81-s + 8·87-s + 8·92-s + 14·100-s − 22·101-s + 44·103-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 1/2·4-s + 2·9-s − 1.15·12-s + 16-s + 0.485·17-s + 1.66·23-s + 14/5·25-s − 0.371·29-s + 36-s + 1.52·43-s − 2.30·48-s − 2.14·49-s − 1.12·51-s + 6.04·53-s − 4.09·61-s + 3/8·64-s + 0.242·68-s − 3.85·69-s − 6.46·75-s + 6.75·79-s − 5/3·81-s + 0.857·87-s + 0.834·92-s + 7/5·100-s − 2.18·101-s + 4.33·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.084033100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.084033100\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + T + T^{2} )^{4} \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - T^{2} - 3 T^{4} + p^{2} T^{6} - p^{2} T^{8} + p^{4} T^{10} - 3 p^{4} T^{12} - p^{6} T^{14} + p^{8} T^{16} \) |
| 5 | \( ( 1 - 7 T^{2} + 24 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( 1 + 15 T^{2} + 109 T^{4} + 270 T^{6} + 30 T^{8} + 270 p^{2} T^{10} + 109 p^{4} T^{12} + 15 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 + 18 T^{2} + 203 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - T - 29 T^{2} + 4 T^{3} + 594 T^{4} + 4 p T^{5} - 29 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 + 24 T^{2} + 322 T^{4} - 11232 T^{6} - 273741 T^{8} - 11232 p^{2} T^{10} + 322 p^{4} T^{12} + 24 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( ( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 + T - 19 T^{2} - 38 T^{3} - 470 T^{4} - 38 p T^{5} - 19 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 115 T^{2} + 5224 T^{4} - 115 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 + 79 T^{2} + 2457 T^{4} + 82634 T^{6} + 3900566 T^{8} + 82634 p^{2} T^{10} + 2457 p^{4} T^{12} + 79 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( 1 + 155 T^{2} + 14661 T^{4} + 930310 T^{6} + 44472710 T^{8} + 930310 p^{2} T^{10} + 14661 p^{4} T^{12} + 155 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 - 5 T - 63 T^{2} - 10 T^{3} + 4820 T^{4} - 10 p T^{5} - 63 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 11 T + 98 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{4} \) |
| 59 | \( 1 + 104 T^{2} + 4482 T^{4} - 65312 T^{6} - 9648301 T^{8} - 65312 p^{2} T^{10} + 4482 p^{4} T^{12} + 104 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 + 16 T + 87 T^{2} + 752 T^{3} + 9224 T^{4} + 752 p T^{5} + 87 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 + 247 T^{2} + 36885 T^{4} + 3741062 T^{6} + 287946854 T^{8} + 3741062 p^{2} T^{10} + 36885 p^{4} T^{12} + 247 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 54 T^{2} - 2125 T^{4} - 54 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 186 T^{2} + 16859 T^{4} - 186 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 15 T + 210 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 248 T^{2} + 27454 T^{4} - 248 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 + 160 T^{2} + 8866 T^{4} + 142720 T^{6} + 14506915 T^{8} + 142720 p^{2} T^{10} + 8866 p^{4} T^{12} + 160 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( 1 + 295 T^{2} + 47169 T^{4} + 6206210 T^{6} + 681153230 T^{8} + 6206210 p^{2} T^{10} + 47169 p^{4} T^{12} + 295 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−4.90590418366153579172877211429, −4.70579396802107896496553137149, −4.70539492821493616775028867546, −4.52822030473387907445217988637, −4.40726333376805503480558811124, −4.02752064850149023721945472630, −3.95328059014769722412436031278, −3.75206209983789581455313749093, −3.67878274951395642826261104006, −3.53828062474881842608722733893, −3.24786289232962283447914017046, −3.13558154817047365711907908753, −3.08573410359696370047508131333, −2.83574566805473002639543772657, −2.78484644899327010629711979436, −2.36259700712080783233910407137, −2.31582799341852914681894039369, −2.16898152478762723623209492597, −1.90694529078833871603736934441, −1.68909613561630295014376903440, −1.15060972794886268533283965689, −1.08197688517649470736827724740, −0.903118071279828179056018451984, −0.857135967625351433750783888250, −0.36701880213898301268550548762,
0.36701880213898301268550548762, 0.857135967625351433750783888250, 0.903118071279828179056018451984, 1.08197688517649470736827724740, 1.15060972794886268533283965689, 1.68909613561630295014376903440, 1.90694529078833871603736934441, 2.16898152478762723623209492597, 2.31582799341852914681894039369, 2.36259700712080783233910407137, 2.78484644899327010629711979436, 2.83574566805473002639543772657, 3.08573410359696370047508131333, 3.13558154817047365711907908753, 3.24786289232962283447914017046, 3.53828062474881842608722733893, 3.67878274951395642826261104006, 3.75206209983789581455313749093, 3.95328059014769722412436031278, 4.02752064850149023721945472630, 4.40726333376805503480558811124, 4.52822030473387907445217988637, 4.70539492821493616775028867546, 4.70579396802107896496553137149, 4.90590418366153579172877211429
Plot not available for L-functions of degree greater than 10.
