| L(s) = 1 | − 5·4-s + 7·16-s − 2·17-s + 4·25-s + 4·29-s + 30·43-s + 36·49-s + 34·53-s − 26·61-s + 14·64-s + 10·68-s + 6·79-s − 20·100-s + 4·101-s + 28·103-s + 26·107-s − 48·113-s − 20·116-s + 25·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 5/2·4-s + 7/4·16-s − 0.485·17-s + 4/5·25-s + 0.742·29-s + 4.57·43-s + 36/7·49-s + 4.67·53-s − 3.32·61-s + 7/4·64-s + 1.21·68-s + 0.675·79-s − 2·100-s + 0.398·101-s + 2.75·103-s + 2.51·107-s − 4.51·113-s − 1.85·116-s + 2.27·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.892514603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.892514603\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 5 T^{2} + 9 p T^{4} + 41 T^{6} + 9 p^{3} T^{8} + 5 p^{4} T^{10} + p^{6} T^{12} \) |
| 5 | \( 1 - 4 T^{2} + 36 T^{4} - 241 T^{6} + 36 p^{2} T^{8} - 4 p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 36 T^{2} + 572 T^{4} - 5165 T^{6} + 572 p^{2} T^{8} - 36 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 25 T^{2} + 485 T^{4} - 5601 T^{6} + 485 p^{2} T^{8} - 25 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 + T + 35 T^{2} + 47 T^{3} + 35 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 19 | \( 1 - 93 T^{2} + 3917 T^{4} - 95369 T^{6} + 3917 p^{2} T^{8} - 93 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 + 20 T^{2} + 91 T^{3} + 20 p T^{4} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 - 2 T + 72 T^{2} - 3 p T^{3} + 72 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 - 12 T^{2} + 824 T^{4} - 48797 T^{6} + 824 p^{2} T^{8} - 12 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( 1 - 56 T^{2} + 4660 T^{4} - 150689 T^{6} + 4660 p^{2} T^{8} - 56 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( 1 - 173 T^{2} + 14457 T^{4} - 740849 T^{6} + 14457 p^{2} T^{8} - 173 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 - 15 T + 176 T^{2} - 1331 T^{3} + 176 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 261 T^{2} + 29285 T^{4} - 1807289 T^{6} + 29285 p^{2} T^{8} - 261 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 - 17 T + 225 T^{2} - 1761 T^{3} + 225 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 - 286 T^{2} + 37671 T^{4} - 2853988 T^{6} + 37671 p^{2} T^{8} - 286 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( ( 1 + 13 T + 167 T^{2} + 1419 T^{3} + 167 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 189 T^{2} + 11465 T^{4} - 439313 T^{6} + 11465 p^{2} T^{8} - 189 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( 1 - 244 T^{2} + 32208 T^{4} - 2788141 T^{6} + 32208 p^{2} T^{8} - 244 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( 1 - 132 T^{2} + 19884 T^{4} - 1422313 T^{6} + 19884 p^{2} T^{8} - 132 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 3 T + 219 T^{2} - 447 T^{3} + 219 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 436 T^{2} + 83400 T^{4} - 8978917 T^{6} + 83400 p^{2} T^{8} - 436 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 - 333 T^{2} + 57485 T^{4} - 6354113 T^{6} + 57485 p^{2} T^{8} - 333 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 + 5 T^{2} + 8710 T^{4} - 929407 T^{6} + 8710 p^{2} T^{8} + 5 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−5.00171811512600499612870916933, −4.65153713391220907124012058918, −4.48569166050070118406325560932, −4.36965224806786746449139638394, −4.36080469016562574588959453430, −4.30613812988014028224377894392, −4.12480374571145881596078109018, −3.96162832640416538757463410137, −3.85093449549676267761320583250, −3.63363874953090742454773412152, −3.53081691537283655566765211294, −3.10880848788406165685559886560, −3.01224409845638604410834341729, −2.72400930474258247374171246681, −2.71658151751720849364412184373, −2.40493117856235023151989027897, −2.17297939242587534594821052711, −2.17104215548601367344215788341, −2.12018199270485525594339372696, −1.37316367862143324126032901249, −1.31306878585220114449510101521, −0.889525709410778723954731107381, −0.74513726900819630907797377474, −0.60581511918470796796317907006, −0.38219061254588575890010683466,
0.38219061254588575890010683466, 0.60581511918470796796317907006, 0.74513726900819630907797377474, 0.889525709410778723954731107381, 1.31306878585220114449510101521, 1.37316367862143324126032901249, 2.12018199270485525594339372696, 2.17104215548601367344215788341, 2.17297939242587534594821052711, 2.40493117856235023151989027897, 2.71658151751720849364412184373, 2.72400930474258247374171246681, 3.01224409845638604410834341729, 3.10880848788406165685559886560, 3.53081691537283655566765211294, 3.63363874953090742454773412152, 3.85093449549676267761320583250, 3.96162832640416538757463410137, 4.12480374571145881596078109018, 4.30613812988014028224377894392, 4.36080469016562574588959453430, 4.36965224806786746449139638394, 4.48569166050070118406325560932, 4.65153713391220907124012058918, 5.00171811512600499612870916933
Plot not available for L-functions of degree greater than 10.
