| L(s) = 1 | − 6·3-s − 3·4-s + 21·9-s + 18·12-s − 6·13-s + 6·16-s + 18·17-s + 18·25-s − 56·27-s + 6·29-s − 63·36-s + 36·39-s + 12·43-s − 36·48-s − 108·51-s + 18·52-s + 18·53-s + 6·61-s − 10·64-s − 54·68-s − 108·75-s − 24·79-s + 126·81-s − 36·87-s − 54·100-s − 36·101-s − 48·103-s + ⋯ |
| L(s) = 1 | − 3.46·3-s − 3/2·4-s + 7·9-s + 5.19·12-s − 1.66·13-s + 3/2·16-s + 4.36·17-s + 18/5·25-s − 10.7·27-s + 1.11·29-s − 10.5·36-s + 5.76·39-s + 1.82·43-s − 5.19·48-s − 15.1·51-s + 2.49·52-s + 2.47·53-s + 0.768·61-s − 5/4·64-s − 6.54·68-s − 12.4·75-s − 2.70·79-s + 14·81-s − 3.85·87-s − 5.39·100-s − 3.58·101-s − 4.72·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.991096424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.991096424\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{3} \) |
| 3 | \( ( 1 + T )^{6} \) |
| 7 | \( 1 \) |
| 13 | \( 1 + 6 T + 27 T^{2} + 132 T^{3} + 27 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 5 | \( 1 - 18 T^{2} + 171 T^{4} - 1056 T^{6} + 171 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 51 T^{2} + 1182 T^{4} - 16315 T^{6} + 1182 p^{2} T^{8} - 51 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 - 9 T + 66 T^{2} - 299 T^{3} + 66 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 19 | \( 1 - 15 T^{2} + 294 T^{4} - 10091 T^{6} + 294 p^{2} T^{8} - 15 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 + 27 T^{2} + 98 T^{3} + 27 p T^{4} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 - 3 T + 42 T^{2} - 251 T^{3} + 42 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{3} \) |
| 37 | \( 1 - 198 T^{2} + 17127 T^{4} - 826576 T^{6} + 17127 p^{2} T^{8} - 198 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( 1 - 138 T^{2} + 207 p T^{4} - 369144 T^{6} + 207 p^{3} T^{8} - 138 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 - 6 T + 93 T^{2} - 412 T^{3} + 93 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 87 T^{2} + 6942 T^{4} - 384019 T^{6} + 6942 p^{2} T^{8} - 87 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 - 3 T + p T^{2} )^{6} \) |
| 59 | \( 1 - 183 T^{2} + 17154 T^{4} - 1116363 T^{6} + 17154 p^{2} T^{8} - 183 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( ( 1 - 3 T + 138 T^{2} - 433 T^{3} + 138 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 207 T^{2} + 23046 T^{4} - 1818475 T^{6} + 23046 p^{2} T^{8} - 207 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( 1 - 375 T^{2} + 61734 T^{4} - 5700859 T^{6} + 61734 p^{2} T^{8} - 375 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( 1 - 114 T^{2} + 19875 T^{4} - 1250368 T^{6} + 19875 p^{2} T^{8} - 114 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 12 T + 189 T^{2} + 1368 T^{3} + 189 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 126 T^{2} + 10791 T^{4} - 486516 T^{6} + 10791 p^{2} T^{8} - 126 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 - 318 T^{2} + 54447 T^{4} - 5892352 T^{6} + 54447 p^{2} T^{8} - 318 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 - 258 T^{2} + 32163 T^{4} - 3209600 T^{6} + 32163 p^{2} T^{8} - 258 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−4.51715104037804041290431066302, −4.28360137221759930476153120405, −4.09019555994001354656451697924, −4.08460192359508801677620501846, −3.95288207494622721976349311613, −3.77088947620707027483228491265, −3.55047174476379153319490029951, −3.53939495321848975571374444581, −3.10687700509933242271827901417, −3.10283259333111955141840565246, −2.95543372924244586567087513202, −2.82799280503278164586166504882, −2.53859065225688592920782705139, −2.44632161441704331401152108386, −2.43496629177153263633309935534, −1.85136676800952173722273364426, −1.79088174590001586094171823003, −1.31804319770800074786640198618, −1.24112192927333231287728538630, −1.18672685047506824546613264301, −1.11363541562416449084375398777, −0.938517883156279260429224591638, −0.47749901798626313146871958183, −0.44261513362655720622708959670, −0.39450295748348364428567178670,
0.39450295748348364428567178670, 0.44261513362655720622708959670, 0.47749901798626313146871958183, 0.938517883156279260429224591638, 1.11363541562416449084375398777, 1.18672685047506824546613264301, 1.24112192927333231287728538630, 1.31804319770800074786640198618, 1.79088174590001586094171823003, 1.85136676800952173722273364426, 2.43496629177153263633309935534, 2.44632161441704331401152108386, 2.53859065225688592920782705139, 2.82799280503278164586166504882, 2.95543372924244586567087513202, 3.10283259333111955141840565246, 3.10687700509933242271827901417, 3.53939495321848975571374444581, 3.55047174476379153319490029951, 3.77088947620707027483228491265, 3.95288207494622721976349311613, 4.08460192359508801677620501846, 4.09019555994001354656451697924, 4.28360137221759930476153120405, 4.51715104037804041290431066302
Plot not available for L-functions of degree greater than 10.
