| L(s) = 1 | − 6·3-s − 3·4-s + 21·9-s + 18·12-s + 6·16-s + 16·17-s − 3·25-s − 56·27-s + 22·29-s − 63·36-s − 24·43-s − 36·48-s + 49-s − 96·51-s + 10·53-s + 44·61-s − 10·64-s − 48·68-s + 18·75-s + 62·79-s + 126·81-s − 132·87-s + 9·100-s + 26·101-s + 2·103-s + 2·107-s + 168·108-s + ⋯ |
| L(s) = 1 | − 3.46·3-s − 3/2·4-s + 7·9-s + 5.19·12-s + 3/2·16-s + 3.88·17-s − 3/5·25-s − 10.7·27-s + 4.08·29-s − 10.5·36-s − 3.65·43-s − 5.19·48-s + 1/7·49-s − 13.4·51-s + 1.37·53-s + 5.63·61-s − 5/4·64-s − 5.82·68-s + 2.07·75-s + 6.97·79-s + 14·81-s − 14.1·87-s + 9/10·100-s + 2.58·101-s + 0.197·103-s + 0.193·107-s + 16.1·108-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.274877442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.274877442\) |
| \(L(\frac{3}{2})\) |
|
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 | \( ( 1 + T )^{6} \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 3 T^{2} + 29 T^{4} + 151 T^{6} + 29 p^{2} T^{8} + 3 p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - T^{2} + 5 T^{4} - 657 T^{6} + 5 p^{2} T^{8} - p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 25 T^{2} + 65 T^{4} + 1959 T^{6} + 65 p^{2} T^{8} - 25 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 - 8 T + 63 T^{2} - 264 T^{3} + 63 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 19 | \( 1 - 34 T^{2} + 871 T^{4} - 18172 T^{6} + 871 p^{2} T^{8} - 34 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 + 41 T^{2} - 56 T^{3} + 41 p T^{4} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 - 11 T + 111 T^{2} - 21 p T^{3} + 111 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 - 61 T^{2} + 2665 T^{4} - 91945 T^{6} + 2665 p^{2} T^{8} - 61 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( 1 - 70 T^{2} + 103 T^{4} + 82796 T^{6} + 103 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( 1 - 114 T^{2} + 8591 T^{4} - 405212 T^{6} + 8591 p^{2} T^{8} - 114 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 + 12 T + 149 T^{2} + 928 T^{3} + 149 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 202 T^{2} + 19631 T^{4} - 1156428 T^{6} + 19631 p^{2} T^{8} - 202 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 - 5 T + 123 T^{2} - 573 T^{3} + 123 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 - 257 T^{2} + 32289 T^{4} - 2403737 T^{6} + 32289 p^{2} T^{8} - 257 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( ( 1 - 22 T + 335 T^{2} - 3012 T^{3} + 335 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 334 T^{2} + 50615 T^{4} - 4374468 T^{6} + 50615 p^{2} T^{8} - 334 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( 1 - 262 T^{2} + 35727 T^{4} - 3147508 T^{6} + 35727 p^{2} T^{8} - 262 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( 1 - 209 T^{2} + 14305 T^{4} - 638873 T^{6} + 14305 p^{2} T^{8} - 209 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 31 T + 513 T^{2} - 5431 T^{3} + 513 p T^{4} - 31 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 73 T^{2} + 19361 T^{4} - 896217 T^{6} + 19361 p^{2} T^{8} - 73 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 - 226 T^{2} + 13871 T^{4} - 295548 T^{6} + 13871 p^{2} T^{8} - 226 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 - 233 T^{2} + 9361 T^{4} + 7759 p T^{6} + 9361 p^{2} T^{8} - 233 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−5.19814380633747278079878816913, −5.11011102041223195861153923535, −5.09976585816706298919058747911, −5.05427421859500622281897805651, −4.76322506326299423218664441644, −4.60480978056679981914173675950, −4.59233512542970971907179389001, −3.95087694424743134395694956402, −3.89706492722386592680517769636, −3.86260748528324846736826221643, −3.74835675718696591972753609541, −3.71988491939334307549913209703, −3.29743633969315272837193817140, −3.19663855314273299053275164368, −2.82895666544067697175718776297, −2.81907475846653145858486374218, −2.17931088873499074046904115102, −2.07633372076128064093763058628, −2.06554270423015911114498770510, −1.39388029542423889152473248458, −1.15204312463816871785117229262, −1.09635942724756829502277685855, −0.790963398533810458004061013533, −0.68467591344960291345838573254, −0.41626363760206870126841240013,
0.41626363760206870126841240013, 0.68467591344960291345838573254, 0.790963398533810458004061013533, 1.09635942724756829502277685855, 1.15204312463816871785117229262, 1.39388029542423889152473248458, 2.06554270423015911114498770510, 2.07633372076128064093763058628, 2.17931088873499074046904115102, 2.81907475846653145858486374218, 2.82895666544067697175718776297, 3.19663855314273299053275164368, 3.29743633969315272837193817140, 3.71988491939334307549913209703, 3.74835675718696591972753609541, 3.86260748528324846736826221643, 3.89706492722386592680517769636, 3.95087694424743134395694956402, 4.59233512542970971907179389001, 4.60480978056679981914173675950, 4.76322506326299423218664441644, 5.05427421859500622281897805651, 5.09976585816706298919058747911, 5.11011102041223195861153923535, 5.19814380633747278079878816913
Plot not available for L-functions of degree greater than 10.
