| L(s) = 1 | + 2-s − 9-s + 2·17-s − 18-s − 2·23-s − 25-s + 2·34-s − 2·41-s − 2·46-s + 2·47-s + 18·49-s − 50-s − 2·73-s + 2·79-s − 2·82-s + 2·94-s + 18·98-s − 121-s + 127-s + 131-s + 137-s + 139-s − 2·146-s + 149-s + 151-s − 2·153-s + 157-s + ⋯ |
| L(s) = 1 | + 2-s − 9-s + 2·17-s − 18-s − 2·23-s − 25-s + 2·34-s − 2·41-s − 2·46-s + 2·47-s + 18·49-s − 50-s − 2·73-s + 2·79-s − 2·82-s + 2·94-s + 18·98-s − 121-s + 127-s + 131-s + 137-s + 139-s − 2·146-s + 149-s + 151-s − 2·153-s + 157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{54} \cdot 359^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{54} \cdot 359^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.240158002\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.240158002\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} \) |
| 359 | \( ( 1 + T )^{18} \) |
| good | 3 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} ) \) |
| 5 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} ) \) |
| 7 | \( ( 1 - T )^{18}( 1 + T )^{18} \) |
| 11 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} ) \) |
| 13 | \( ( 1 + T^{2} )^{18} \) |
| 17 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} )^{2} \) |
| 19 | \( ( 1 + T^{2} )^{18} \) |
| 23 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} )^{2} \) |
| 29 | \( ( 1 + T^{2} )^{18} \) |
| 31 | \( ( 1 - T )^{18}( 1 + T )^{18} \) |
| 37 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} ) \) |
| 41 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} )^{2} \) |
| 43 | \( ( 1 + T^{2} )^{18} \) |
| 47 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} )^{2} \) |
| 53 | \( ( 1 + T^{2} )^{18} \) |
| 59 | \( ( 1 + T^{2} )^{18} \) |
| 61 | \( ( 1 + T^{2} )^{18} \) |
| 67 | \( ( 1 + T^{2} )^{18} \) |
| 71 | \( ( 1 - T )^{18}( 1 + T )^{18} \) |
| 73 | \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} + T^{15} + T^{16} + T^{17} + T^{18} )^{2} \) |
| 79 | \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{15} + T^{16} - T^{17} + T^{18} )^{2} \) |
| 83 | \( ( 1 + T^{2} )^{18} \) |
| 89 | \( ( 1 - T )^{18}( 1 + T )^{18} \) |
| 97 | \( ( 1 - T )^{18}( 1 + T )^{18} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{36} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.24873905499384032082559976584, −2.24685265968568588675226232508, −2.23878208071234479440499139984, −2.21951451281488399377023570658, −2.15774378866898994505230756276, −1.96474740131150746514095683532, −1.79453973012996529124245118818, −1.69991618003679776808011601461, −1.65194485585104052133900903630, −1.60258878383849255843666619005, −1.56842557644819776551949553840, −1.47701595381855296834426841211, −1.47262763283886738136534785202, −1.40429590200286120261381336293, −1.17158710316032038132105881279, −1.13800666926895219897710244748, −1.11472902265782623534808621231, −0.996926147869780284629223659688, −0.990336949709210884526503798794, −0.886390957637415641749754806031, −0.799330076475521090355335286083, −0.789291251035774917778280643853, −0.70622560025376278101023908869, −0.50347692938117134674454567475, −0.19255904285134469563879761273,
0.19255904285134469563879761273, 0.50347692938117134674454567475, 0.70622560025376278101023908869, 0.789291251035774917778280643853, 0.799330076475521090355335286083, 0.886390957637415641749754806031, 0.990336949709210884526503798794, 0.996926147869780284629223659688, 1.11472902265782623534808621231, 1.13800666926895219897710244748, 1.17158710316032038132105881279, 1.40429590200286120261381336293, 1.47262763283886738136534785202, 1.47701595381855296834426841211, 1.56842557644819776551949553840, 1.60258878383849255843666619005, 1.65194485585104052133900903630, 1.69991618003679776808011601461, 1.79453973012996529124245118818, 1.96474740131150746514095683532, 2.15774378866898994505230756276, 2.21951451281488399377023570658, 2.23878208071234479440499139984, 2.24685265968568588675226232508, 2.24873905499384032082559976584
Plot not available for L-functions of degree greater than 10.
