| L(s) = 1 | + 6·3-s − 3·4-s + 21·9-s − 18·12-s + 6·16-s − 24·17-s − 32·23-s + 13·25-s + 56·27-s + 26·29-s − 63·36-s + 16·43-s + 36·48-s + 25·49-s − 144·51-s + 30·53-s − 20·61-s − 10·64-s + 72·68-s − 192·69-s + 78·75-s − 10·79-s + 126·81-s + 156·87-s + 96·92-s − 39·100-s − 10·101-s + ⋯ |
| L(s) = 1 | + 3.46·3-s − 3/2·4-s + 7·9-s − 5.19·12-s + 3/2·16-s − 5.82·17-s − 6.67·23-s + 13/5·25-s + 10.7·27-s + 4.82·29-s − 10.5·36-s + 2.43·43-s + 5.19·48-s + 25/7·49-s − 20.1·51-s + 4.12·53-s − 2.56·61-s − 5/4·64-s + 8.73·68-s − 23.1·69-s + 9.00·75-s − 1.12·79-s + 14·81-s + 16.7·87-s + 10.0·92-s − 3.89·100-s − 0.995·101-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
| \(L(1)\) |
\(\approx\) |
\(12.52818718\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.52818718\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 13 T^{2} + 9 p T^{4} - 49 T^{6} + 9 p^{3} T^{8} - 13 p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 25 T^{2} + 269 T^{4} - 2001 T^{6} + 269 p^{2} T^{8} - 25 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 3 p T^{2} + 593 T^{4} - 7553 T^{6} + 593 p^{2} T^{8} - 3 p^{5} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 + 12 T + 71 T^{2} + 304 T^{3} + 71 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 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 | \( ( 1 + 16 T + 145 T^{2} + 840 T^{3} + 145 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 - 13 T + 99 T^{2} - 531 T^{3} + 99 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 - 61 T^{2} - 79 T^{4} + 63399 T^{6} - 79 p^{2} T^{8} - 61 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( 1 - 118 T^{2} + 5351 T^{4} - 174132 T^{6} + 5351 p^{2} T^{8} - 118 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( 1 - 162 T^{2} + 13007 T^{4} - 656636 T^{6} + 13007 p^{2} T^{8} - 162 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 - 8 T + 85 T^{2} - 8 p T^{3} + 85 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 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 | \( ( 1 - 15 T + 87 T^{2} - 343 T^{3} + 87 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 - 313 T^{2} + 42705 T^{4} - 3270673 T^{6} + 42705 p^{2} T^{8} - 313 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( ( 1 + 10 T + 207 T^{2} + 1228 T^{3} + 207 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 + 2 T^{2} + 6263 T^{4} - 222180 T^{6} + 6263 p^{2} T^{8} + 2 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( 1 - 246 T^{2} + 30927 T^{4} - 2616468 T^{6} + 30927 p^{2} T^{8} - 246 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( 1 - 369 T^{2} + 60401 T^{4} - 5663609 T^{6} + 60401 p^{2} T^{8} - 369 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 5 T + 33 T^{2} - 679 T^{3} + 33 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - T^{2} + 9185 T^{4} - 472977 T^{6} + 9185 p^{2} T^{8} - p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 - 482 T^{2} + 100719 T^{4} - 11702012 T^{6} + 100719 p^{2} T^{8} - 482 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 - 505 T^{2} + 111553 T^{4} - 13963489 T^{6} + 111553 p^{2} T^{8} - 505 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−4.97265201690003163443569464267, −4.93932148335929831596965714903, −4.63236667955651995261927928729, −4.48838879773782674869707215619, −4.44576254840836318438154960731, −4.41410185975906623269481760022, −4.25133035660312930434174265402, −4.24182289020416240147806044121, −3.97449162334367827988991945558, −3.82253387853839169425188470607, −3.77955172827098274832144802424, −3.40510958822749639458182170649, −3.07963966986762259070719656312, −3.04885341756507207720726630422, −2.61392743240923314293936447157, −2.59293991548392651737847203252, −2.54203323910779082045144166718, −2.22076482719703413973235303162, −2.13397469367192747351334117366, −2.06510733659147620384261976194, −1.77674891553454029816706996252, −1.51694862897657176102182033136, −0.818463950584149866922661104069, −0.64813531926036361537425921693, −0.53068843562592457290458618166,
0.53068843562592457290458618166, 0.64813531926036361537425921693, 0.818463950584149866922661104069, 1.51694862897657176102182033136, 1.77674891553454029816706996252, 2.06510733659147620384261976194, 2.13397469367192747351334117366, 2.22076482719703413973235303162, 2.54203323910779082045144166718, 2.59293991548392651737847203252, 2.61392743240923314293936447157, 3.04885341756507207720726630422, 3.07963966986762259070719656312, 3.40510958822749639458182170649, 3.77955172827098274832144802424, 3.82253387853839169425188470607, 3.97449162334367827988991945558, 4.24182289020416240147806044121, 4.25133035660312930434174265402, 4.41410185975906623269481760022, 4.44576254840836318438154960731, 4.48838879773782674869707215619, 4.63236667955651995261927928729, 4.93932148335929831596965714903, 4.97265201690003163443569464267
Plot not available for L-functions of degree greater than 10.
