| L(s) = 1 | + 4·4-s + 2·7-s + 2·9-s − 2·13-s + 6·16-s − 4·25-s + 8·28-s + 8·36-s + 3·49-s − 8·52-s + 4·63-s + 4·64-s − 2·67-s + 81-s − 4·91-s − 16·100-s + 2·103-s − 2·109-s + 12·112-s − 4·117-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 12·144-s + 149-s + ⋯ |
| L(s) = 1 | + 4·4-s + 2·7-s + 2·9-s − 2·13-s + 6·16-s − 4·25-s + 8·28-s + 8·36-s + 3·49-s − 8·52-s + 4·63-s + 4·64-s − 2·67-s + 81-s − 4·91-s − 16·100-s + 2·103-s − 2·109-s + 12·112-s − 4·117-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 12·144-s + 149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 29^{48}\right)^{s/2} \, \Gamma_{\C}(s)^{24} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 29^{48}\right)^{s/2} \, \Gamma_{\C}(s)^{24} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(6.101315168\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.101315168\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2} \) |
| 29 | \( 1 \) |
| good | 2 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{4} \) |
| 5 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{4} \) |
| 7 | \( ( 1 - T + T^{5} - T^{6} + T^{7} - T^{8} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{16} + T^{17} - T^{18} + T^{19} - T^{23} + T^{24} )^{2} \) |
| 11 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{4} \) |
| 13 | \( ( 1 + T - T^{5} - T^{6} - T^{7} - T^{8} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} - T^{16} - T^{17} - T^{18} - T^{19} + T^{23} + T^{24} )^{2} \) |
| 17 | \( ( 1 + T^{2} )^{24} \) |
| 19 | \( 1 + T^{2} - T^{10} - T^{12} - T^{14} - T^{16} + T^{20} + T^{22} + T^{24} + T^{26} + T^{28} - T^{32} - T^{34} - T^{36} - T^{38} + T^{46} + T^{48} \) |
| 23 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{4} \) |
| 31 | \( 1 + T^{2} - T^{10} - T^{12} - T^{14} - T^{16} + T^{20} + T^{22} + T^{24} + T^{26} + T^{28} - T^{32} - T^{34} - T^{36} - T^{38} + T^{46} + T^{48} \) |
| 37 | \( 1 + T^{2} - T^{10} - T^{12} - T^{14} - T^{16} + T^{20} + T^{22} + T^{24} + T^{26} + T^{28} - T^{32} - T^{34} - T^{36} - T^{38} + T^{46} + T^{48} \) |
| 41 | \( ( 1 + T^{2} )^{24} \) |
| 43 | \( 1 + T^{2} - T^{10} - T^{12} - T^{14} - T^{16} + T^{20} + T^{22} + T^{24} + T^{26} + T^{28} - T^{32} - T^{34} - T^{36} - T^{38} + T^{46} + T^{48} \) |
| 47 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{4} \) |
| 53 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{4} \) |
| 59 | \( ( 1 - T )^{24}( 1 + T )^{24} \) |
| 61 | \( 1 + T^{2} - T^{10} - T^{12} - T^{14} - T^{16} + T^{20} + T^{22} + T^{24} + T^{26} + T^{28} - T^{32} - T^{34} - T^{36} - T^{38} + T^{46} + T^{48} \) |
| 67 | \( ( 1 + T - T^{5} - T^{6} - T^{7} - T^{8} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} - T^{16} - T^{17} - T^{18} - T^{19} + T^{23} + T^{24} )^{2} \) |
| 71 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{4} \) |
| 73 | \( 1 + T^{2} - T^{10} - T^{12} - T^{14} - T^{16} + T^{20} + T^{22} + T^{24} + T^{26} + T^{28} - T^{32} - T^{34} - T^{36} - T^{38} + T^{46} + T^{48} \) |
| 79 | \( 1 + T^{2} - T^{10} - T^{12} - T^{14} - T^{16} + T^{20} + T^{22} + T^{24} + T^{26} + T^{28} - T^{32} - T^{34} - T^{36} - T^{38} + T^{46} + T^{48} \) |
| 83 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{4} \) |
| 89 | \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{4} \) |
| 97 | \( 1 + T^{2} - T^{10} - T^{12} - T^{14} - T^{16} + T^{20} + T^{22} + T^{24} + T^{26} + T^{28} - T^{32} - T^{34} - T^{36} - T^{38} + T^{46} + T^{48} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{48} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−1.99807705137150253941996510453, −1.95655758618763198208599369894, −1.88415597651392593574899784920, −1.71639290035704922173673165775, −1.70061124788177607234121540452, −1.58068628289035731063943644014, −1.55985506045017868383519535106, −1.51110283172942226236013210585, −1.51101839856269378776781649414, −1.42468404928078472860675137237, −1.41706068632629056491045422042, −1.35371139107904117761525929787, −1.31644057167563678991335379583, −1.31016054440299041107434375314, −1.17430637657626183448384312927, −1.10318329860255211215828082534, −1.06639215672873844562521664396, −1.01006827646402410802868161425, −0.907283815424059814689246507850, −0.800422261972713066664649704950, −0.797817780164104121578615914482, −0.66185113984262191023502915910, −0.58662283106179555050582212728, −0.40070512339981147031590914024, −0.23579361826986376441570693428,
0.23579361826986376441570693428, 0.40070512339981147031590914024, 0.58662283106179555050582212728, 0.66185113984262191023502915910, 0.797817780164104121578615914482, 0.800422261972713066664649704950, 0.907283815424059814689246507850, 1.01006827646402410802868161425, 1.06639215672873844562521664396, 1.10318329860255211215828082534, 1.17430637657626183448384312927, 1.31016054440299041107434375314, 1.31644057167563678991335379583, 1.35371139107904117761525929787, 1.41706068632629056491045422042, 1.42468404928078472860675137237, 1.51101839856269378776781649414, 1.51110283172942226236013210585, 1.55985506045017868383519535106, 1.58068628289035731063943644014, 1.70061124788177607234121540452, 1.71639290035704922173673165775, 1.88415597651392593574899784920, 1.95655758618763198208599369894, 1.99807705137150253941996510453
Plot not available for L-functions of degree greater than 10.
