| L(s) = 1 | − 6·3-s + 21·9-s − 2·13-s − 4·23-s + 10·25-s − 56·27-s − 4·29-s + 12·39-s + 32·43-s − 3·49-s − 36·53-s + 28·61-s + 24·69-s − 60·75-s + 32·79-s + 126·81-s + 24·87-s − 24·101-s − 32·103-s − 12·107-s + 20·113-s − 42·117-s + 22·121-s + 127-s − 192·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 3.46·3-s + 7·9-s − 0.554·13-s − 0.834·23-s + 2·25-s − 10.7·27-s − 0.742·29-s + 1.92·39-s + 4.87·43-s − 3/7·49-s − 4.94·53-s + 3.58·61-s + 2.88·69-s − 6.92·75-s + 3.60·79-s + 14·81-s + 2.57·87-s − 2.38·101-s − 3.15·103-s − 1.16·107-s + 1.88·113-s − 3.88·117-s + 2·121-s + 0.0887·127-s − 16.9·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7840545054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7840545054\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T )^{6} \) |
| 7 | \( ( 1 + T^{2} )^{3} \) |
| 13 | \( 1 + 2 T + 27 T^{2} + 44 T^{3} + 27 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 5 | \( 1 - 2 p T^{2} + 23 T^{4} + 36 T^{6} + 23 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 2 p T^{2} + 263 T^{4} - 2724 T^{6} + 263 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 + 35 T^{2} + 16 T^{3} + 35 p T^{4} + p^{3} T^{6} )^{2} \) |
| 19 | \( 1 - 50 T^{2} + 1063 T^{4} - 16988 T^{6} + 1063 p^{2} T^{8} - 50 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 + 2 T + 49 T^{2} + 84 T^{3} + 49 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 + 2 T + 35 T^{2} + 76 T^{3} + 35 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 - 10 T^{2} + 17 p T^{4} + 29364 T^{6} + 17 p^{3} T^{8} - 10 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( 1 - 174 T^{2} + 13943 T^{4} - 655652 T^{6} + 13943 p^{2} T^{8} - 174 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( 1 - 114 T^{2} + 4031 T^{4} - 84716 T^{6} + 4031 p^{2} T^{8} - 114 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 - 16 T + 161 T^{2} - 1248 T^{3} + 161 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 226 T^{2} + 23391 T^{4} - 1405228 T^{6} + 23391 p^{2} T^{8} - 226 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 + 6 T + p T^{2} )^{6} \) |
| 59 | \( 1 - 314 T^{2} + 42855 T^{4} - 3281564 T^{6} + 42855 p^{2} T^{8} - 314 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( ( 1 - 14 T + 11 T^{2} + 684 T^{3} + 11 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 274 T^{2} + 36103 T^{4} - 2978332 T^{6} + 36103 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( 1 - 94 T^{2} + 12111 T^{4} - 940276 T^{6} + 12111 p^{2} T^{8} - 94 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( 1 - 166 T^{2} + 7231 T^{4} - 21268 T^{6} + 7231 p^{2} T^{8} - 166 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 16 T + 309 T^{2} - 2608 T^{3} + 309 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 90 T^{2} + 17495 T^{4} - 1240220 T^{6} + 17495 p^{2} T^{8} - 90 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 - 322 T^{2} + 50175 T^{4} - 5234188 T^{6} + 50175 p^{2} T^{8} - 322 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 - 550 T^{2} + 128975 T^{4} - 16496340 T^{6} + 128975 p^{2} T^{8} - 550 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−4.43636078370550383974812972733, −4.42094822753679360669339917159, −4.12223338912487838863877819826, −3.82693399667821429491334206464, −3.79688872754578418594510120229, −3.67348197838188564036176441413, −3.54184261611085290789808549306, −3.46045356425964481059433798558, −3.30430087422125522506634873751, −2.91893346953572569181758270978, −2.75339550684674185658178231113, −2.55519339048367550310426472584, −2.42637069234976860103099102607, −2.40680976541065475822602636409, −2.22604909628947878518902913316, −2.11996741736345741694370347192, −1.60017890299297787081969672195, −1.43533506043107935826415030398, −1.43382027322472992792863302116, −1.22221734115370871575007070507, −1.13098740940648607438933020389, −0.76023250447179627288217736591, −0.63429076654966409130231620591, −0.26736882370630471092962832615, −0.25813706167670251128205131677,
0.25813706167670251128205131677, 0.26736882370630471092962832615, 0.63429076654966409130231620591, 0.76023250447179627288217736591, 1.13098740940648607438933020389, 1.22221734115370871575007070507, 1.43382027322472992792863302116, 1.43533506043107935826415030398, 1.60017890299297787081969672195, 2.11996741736345741694370347192, 2.22604909628947878518902913316, 2.40680976541065475822602636409, 2.42637069234976860103099102607, 2.55519339048367550310426472584, 2.75339550684674185658178231113, 2.91893346953572569181758270978, 3.30430087422125522506634873751, 3.46045356425964481059433798558, 3.54184261611085290789808549306, 3.67348197838188564036176441413, 3.79688872754578418594510120229, 3.82693399667821429491334206464, 4.12223338912487838863877819826, 4.42094822753679360669339917159, 4.43636078370550383974812972733
Plot not available for L-functions of degree greater than 10.
