| L(s) = 1 | − 4·2-s − 3-s + 8·4-s + 4·6-s − 10·8-s − 2·9-s + 20·11-s − 8·12-s + 8·18-s + 4·19-s − 80·22-s + 18·23-s + 10·24-s + 13·25-s + 3·27-s + 28·32-s − 20·33-s − 16·36-s − 16·38-s + 160·44-s − 72·46-s − 46·47-s − 34·49-s − 52·50-s − 12·54-s − 4·57-s + 10·59-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 0.577·3-s + 4·4-s + 1.63·6-s − 3.53·8-s − 2/3·9-s + 6.03·11-s − 2.30·12-s + 1.88·18-s + 0.917·19-s − 17.0·22-s + 3.75·23-s + 2.04·24-s + 13/5·25-s + 0.577·27-s + 4.94·32-s − 3.48·33-s − 8/3·36-s − 2.59·38-s + 24.1·44-s − 10.6·46-s − 6.70·47-s − 4.85·49-s − 7.35·50-s − 1.63·54-s − 0.529·57-s + 1.30·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 59^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 59^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4249438038\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4249438038\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T + p T^{2} + 2 T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | \( 1 - 10 T + 85 T^{2} - 732 T^{3} + 85 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 2 | \( ( 1 + p T + p T^{2} + T^{3} + p^{2} T^{4} + p^{3} T^{5} + p^{3} T^{6} )^{2} \) |
| 5 | \( 1 - 13 T^{2} + 106 T^{4} - 637 T^{6} + 106 p^{2} T^{8} - 13 p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( ( 1 + 17 T^{2} + T^{3} + 17 p T^{4} + p^{3} T^{6} )^{2} \) |
| 11 | \( ( 1 - 10 T + 60 T^{2} - 234 T^{3} + 60 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 20 T^{2} + 516 T^{4} - 6542 T^{6} + 516 p^{2} T^{8} - 20 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 75 T^{2} + 2696 T^{4} - 57772 T^{6} + 2696 p^{2} T^{8} - 75 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 - 2 T + 31 T^{2} - 113 T^{3} + 31 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( ( 1 - 9 T + 4 p T^{2} - 428 T^{3} + 4 p^{2} T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( 1 - 120 T^{2} + 6965 T^{4} - 249781 T^{6} + 6965 p^{2} T^{8} - 120 p^{4} T^{10} + p^{6} T^{12} \) |
| 31 | \( 1 - 71 T^{2} + 3744 T^{4} - 137216 T^{6} + 3744 p^{2} T^{8} - 71 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( 1 + T^{2} + 1068 T^{4} + 37816 T^{6} + 1068 p^{2} T^{8} + p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( 1 - 188 T^{2} + 15985 T^{4} - 816533 T^{6} + 15985 p^{2} T^{8} - 188 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( 1 - 52 T^{2} + 5868 T^{4} - 182854 T^{6} + 5868 p^{2} T^{8} - 52 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( ( 1 + 23 T + 312 T^{2} + 2568 T^{3} + 312 p T^{4} + 23 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 53 | \( 1 - 197 T^{2} + 19498 T^{4} - 1253909 T^{6} + 19498 p^{2} T^{8} - 197 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( 1 - 167 T^{2} + 17204 T^{4} - 1305528 T^{6} + 17204 p^{2} T^{8} - 167 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( 1 - 344 T^{2} + 52788 T^{4} - 4582178 T^{6} + 52788 p^{2} T^{8} - 344 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( 1 - 120 T^{2} + 132 T^{4} + 542342 T^{6} + 132 p^{2} T^{8} - 120 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( 1 - 191 T^{2} + 21072 T^{4} - 1746740 T^{6} + 21072 p^{2} T^{8} - 191 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 7 T + 152 T^{2} + 759 T^{3} + 152 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( ( 1 - 25 T + 400 T^{2} - 4164 T^{3} + 400 p T^{4} - 25 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( ( 1 + 13 T + 242 T^{2} + 1796 T^{3} + 242 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 240 T^{2} + 15788 T^{4} - 353866 T^{6} + 15788 p^{2} T^{8} - 240 p^{4} T^{10} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.91603822781302418863104118942, −6.82511365034063810918415631404, −6.73918470219920389501117260155, −6.69370075700139909522987664644, −6.43489512827673301310510265015, −6.41076694317030642217371086399, −6.27427348414893496140922450500, −5.97089529695821154609300320714, −5.23344888611309216641120488990, −5.10532225802669375944002241310, −4.96340666105419233682169571608, −4.91737230872597404630533303827, −4.80916649247499712683415799112, −4.07201939147727172577406512633, −4.06977282938096059805462890001, −3.99987280361672570519568626016, −3.25349456703417269689273796349, −3.11793327663736736822782404348, −3.07222543395899032685408247983, −3.01009819221546976033602332409, −1.90169836458706514322098136145, −1.56974727555100750332175828155, −1.55713254441623194389028352354, −1.22245333704650313101855834760, −0.75622189551127714428786083041,
0.75622189551127714428786083041, 1.22245333704650313101855834760, 1.55713254441623194389028352354, 1.56974727555100750332175828155, 1.90169836458706514322098136145, 3.01009819221546976033602332409, 3.07222543395899032685408247983, 3.11793327663736736822782404348, 3.25349456703417269689273796349, 3.99987280361672570519568626016, 4.06977282938096059805462890001, 4.07201939147727172577406512633, 4.80916649247499712683415799112, 4.91737230872597404630533303827, 4.96340666105419233682169571608, 5.10532225802669375944002241310, 5.23344888611309216641120488990, 5.97089529695821154609300320714, 6.27427348414893496140922450500, 6.41076694317030642217371086399, 6.43489512827673301310510265015, 6.69370075700139909522987664644, 6.73918470219920389501117260155, 6.82511365034063810918415631404, 6.91603822781302418863104118942
Plot not available for L-functions of degree greater than 10.
