| L(s) = 1 | + 14·11-s + 8·19-s − 5·25-s + 6·29-s − 20·31-s − 36·41-s − 3·49-s − 12·59-s + 48·61-s + 8·71-s + 34·79-s + 8·101-s − 10·109-s + 65·121-s + 8·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 9·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 4.22·11-s + 1.83·19-s − 25-s + 1.11·29-s − 3.59·31-s − 5.62·41-s − 3/7·49-s − 1.56·59-s + 6.14·61-s + 0.949·71-s + 3.82·79-s + 0.796·101-s − 0.957·109-s + 5.90·121-s + 0.715·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 9/13·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.356299097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.356299097\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + p T^{2} - 8 T^{3} + p^{2} T^{4} + p^{3} T^{6} \) |
| 7 | \( ( 1 + T^{2} )^{3} \) |
| good | 11 | \( ( 1 - 7 T + 41 T^{2} - 146 T^{3} + 41 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 9 T^{2} + 491 T^{4} - 2882 T^{6} + 491 p^{2} T^{8} - 9 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 53 T^{2} + 1539 T^{4} - 31118 T^{6} + 1539 p^{2} T^{8} - 53 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 - 4 T + 43 T^{2} - 160 T^{3} + 43 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 2 p T^{2} + 1887 T^{4} - 43972 T^{6} + 1887 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 3 T + 15 T^{2} - 66 T^{3} + 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 + 10 T + 101 T^{2} + 540 T^{3} + 101 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{3} \) |
| 41 | \( ( 1 + 18 T + 191 T^{2} + 1388 T^{3} + 191 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 178 T^{2} + 15575 T^{4} - 836124 T^{6} + 15575 p^{2} T^{8} - 178 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 105 T^{2} + 6339 T^{4} - 285798 T^{6} + 6339 p^{2} T^{8} - 105 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 130 T^{2} + 13655 T^{4} - 792060 T^{6} + 13655 p^{2} T^{8} - 130 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 6 T + 99 T^{2} + 752 T^{3} + 99 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 - 24 T + 365 T^{2} - 3368 T^{3} + 365 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 174 T^{2} + 20567 T^{4} - 1533188 T^{6} + 20567 p^{2} T^{8} - 174 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 4 T + 193 T^{2} - 504 T^{3} + 193 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 346 T^{2} + 55487 T^{4} - 5172972 T^{6} + 55487 p^{2} T^{8} - 346 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 17 T + 205 T^{2} - 2138 T^{3} + 205 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 70 T^{2} + 1179 T^{4} + 304148 T^{6} + 1179 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 95 T^{2} - 464 T^{3} + 95 p T^{4} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 469 T^{2} + 98339 T^{4} - 12075534 T^{6} + 98339 p^{2} T^{8} - 469 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−4.11878493890318628898420547786, −3.80299912107623642997805773378, −3.76860237057572529255416818438, −3.75413051061583963924056282823, −3.68704531189072962434441378059, −3.65234984488424152752519593998, −3.54584351685850532764271751792, −3.38260032969613950834882027551, −3.25192453055287882380888737702, −3.05790242041639887636274229613, −2.72341613734702548527403455845, −2.56419543747605472563613340376, −2.54839530890903612329051783681, −2.19712290563677367933835013702, −1.99899844331126215284848332938, −1.99818945507953992897763840524, −1.72698415759832084652201745141, −1.66290796688459066011322178459, −1.49204392923838265294462345417, −1.28776140359962260374255744011, −1.13581542292539848872261544992, −0.949453857206305336874816334447, −0.795447834372738572738091558766, −0.44717648594909833894071467562, −0.087242946155818872029956470964,
0.087242946155818872029956470964, 0.44717648594909833894071467562, 0.795447834372738572738091558766, 0.949453857206305336874816334447, 1.13581542292539848872261544992, 1.28776140359962260374255744011, 1.49204392923838265294462345417, 1.66290796688459066011322178459, 1.72698415759832084652201745141, 1.99818945507953992897763840524, 1.99899844331126215284848332938, 2.19712290563677367933835013702, 2.54839530890903612329051783681, 2.56419543747605472563613340376, 2.72341613734702548527403455845, 3.05790242041639887636274229613, 3.25192453055287882380888737702, 3.38260032969613950834882027551, 3.54584351685850532764271751792, 3.65234984488424152752519593998, 3.68704531189072962434441378059, 3.75413051061583963924056282823, 3.76860237057572529255416818438, 3.80299912107623642997805773378, 4.11878493890318628898420547786
Plot not available for L-functions of degree greater than 10.
