| L(s) = 1 | + 4·4-s + 26·13-s + 3·16-s + 26·19-s + 22·25-s + 22·31-s + 146·37-s + 54·43-s + 104·52-s − 136·61-s + 46·64-s − 2·67-s − 482·73-s + 104·76-s − 42·79-s − 284·97-s + 88·100-s + 234·103-s − 130·109-s + 514·121-s + 88·124-s + 127-s + 131-s + 137-s + 139-s + 584·148-s + 149-s + ⋯ |
| L(s) = 1 | + 4-s + 2·13-s + 3/16·16-s + 1.36·19-s + 0.879·25-s + 0.709·31-s + 3.94·37-s + 1.25·43-s + 2·52-s − 2.22·61-s + 0.718·64-s − 0.0298·67-s − 6.60·73-s + 1.36·76-s − 0.531·79-s − 2.92·97-s + 0.879·100-s + 2.27·103-s − 1.19·109-s + 4.24·121-s + 0.709·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 3.94·148-s + 0.00671·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.045815270\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.045815270\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - p^{2} T^{2} + 13 T^{4} - 43 p T^{6} + 13 p^{4} T^{8} - p^{10} T^{10} + p^{12} T^{12} \) |
| 5 | \( 1 - 22 T^{2} + 1132 T^{4} - 17132 T^{6} + 1132 p^{4} T^{8} - 22 p^{8} T^{10} + p^{12} T^{12} \) |
| 11 | \( 1 - 514 T^{2} + 129484 T^{4} - 19636340 T^{6} + 129484 p^{4} T^{8} - 514 p^{8} T^{10} + p^{12} T^{12} \) |
| 13 | \( ( 1 - p T + 269 T^{2} - 1552 T^{3} + 269 p^{2} T^{4} - p^{5} T^{5} + p^{6} T^{6} )^{2} \) |
| 17 | \( 1 - 738 T^{2} + 399555 T^{4} - 131047364 T^{6} + 399555 p^{4} T^{8} - 738 p^{8} T^{10} + p^{12} T^{12} \) |
| 19 | \( ( 1 - 13 T + 845 T^{2} - 6544 T^{3} + 845 p^{2} T^{4} - 13 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 23 | \( 1 - 1890 T^{2} + 1918755 T^{4} - 1249145156 T^{6} + 1918755 p^{4} T^{8} - 1890 p^{8} T^{10} + p^{12} T^{12} \) |
| 29 | \( 1 - 2920 T^{2} + 4462552 T^{4} - 4500431378 T^{6} + 4462552 p^{4} T^{8} - 2920 p^{8} T^{10} + p^{12} T^{12} \) |
| 31 | \( ( 1 - 11 T + 1672 T^{2} - 31873 T^{3} + 1672 p^{2} T^{4} - 11 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 37 | \( ( 1 - 73 T + 5101 T^{2} - 186980 T^{3} + 5101 p^{2} T^{4} - 73 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 41 | \( 1 - 6892 T^{2} + 23337223 T^{4} - 49003439192 T^{6} + 23337223 p^{4} T^{8} - 6892 p^{8} T^{10} + p^{12} T^{12} \) |
| 43 | \( ( 1 - 27 T + 4581 T^{2} - 73568 T^{3} + 4581 p^{2} T^{4} - 27 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 47 | \( 1 - 5616 T^{2} + 20345283 T^{4} - 53758551008 T^{6} + 20345283 p^{4} T^{8} - 5616 p^{8} T^{10} + p^{12} T^{12} \) |
| 53 | \( 1 - 10584 T^{2} + 54376488 T^{4} - 182665815122 T^{6} + 54376488 p^{4} T^{8} - 10584 p^{8} T^{10} + p^{12} T^{12} \) |
| 59 | \( 1 - 12708 T^{2} + 73521072 T^{4} - 288183912842 T^{6} + 73521072 p^{4} T^{8} - 12708 p^{8} T^{10} + p^{12} T^{12} \) |
| 61 | \( ( 1 + 68 T + 8993 T^{2} + 389240 T^{3} + 8993 p^{2} T^{4} + 68 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 67 | \( ( 1 + T + 12563 T^{2} + 4546 T^{3} + 12563 p^{2} T^{4} + p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 71 | \( 1 - 3984 T^{2} + 20461731 T^{4} - 164780765984 T^{6} + 20461731 p^{4} T^{8} - 3984 p^{8} T^{10} + p^{12} T^{12} \) |
| 73 | \( ( 1 + 241 T + 26347 T^{2} + 2035850 T^{3} + 26347 p^{2} T^{4} + 241 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 79 | \( ( 1 + 21 T + 3912 T^{2} + 849823 T^{3} + 3912 p^{2} T^{4} + 21 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 83 | \( 1 - 21858 T^{2} + 183628284 T^{4} - 1109191776932 T^{6} + 183628284 p^{4} T^{8} - 21858 p^{8} T^{10} + p^{12} T^{12} \) |
| 89 | \( 1 - 36766 T^{2} + 634804639 T^{4} - 6402279105860 T^{6} + 634804639 p^{4} T^{8} - 36766 p^{8} T^{10} + p^{12} T^{12} \) |
| 97 | \( ( 1 + 142 T + 22732 T^{2} + 2546324 T^{3} + 22732 p^{2} T^{4} + 142 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
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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
−5.94052238633313809771556097036, −5.71768062725796861053797817553, −5.54625134389718110930114170439, −5.26988169023922264211005098194, −5.10886206414266296204718308278, −4.72668076145032058955664101710, −4.64436320243174583554325373646, −4.49745244407630669016940497275, −4.28213451876345529342830237904, −4.13685929956485157325550663977, −4.00721338089897818726468216057, −3.54787223134188167299788383744, −3.53299566088482406355664034086, −3.11929711110769671265959760439, −3.09953670633459474159964318682, −2.71531738975489791273395354739, −2.70229997024550050459131546638, −2.50361171775021393641953717868, −2.25874812635861677827834267005, −1.67878200857151953617743546971, −1.51608250158679403281508040891, −1.25042611920923603496110497709, −0.988325584629522036154894885596, −0.973022741650039729341429387768, −0.20757162186136360548313347927,
0.20757162186136360548313347927, 0.973022741650039729341429387768, 0.988325584629522036154894885596, 1.25042611920923603496110497709, 1.51608250158679403281508040891, 1.67878200857151953617743546971, 2.25874812635861677827834267005, 2.50361171775021393641953717868, 2.70229997024550050459131546638, 2.71531738975489791273395354739, 3.09953670633459474159964318682, 3.11929711110769671265959760439, 3.53299566088482406355664034086, 3.54787223134188167299788383744, 4.00721338089897818726468216057, 4.13685929956485157325550663977, 4.28213451876345529342830237904, 4.49745244407630669016940497275, 4.64436320243174583554325373646, 4.72668076145032058955664101710, 5.10886206414266296204718308278, 5.26988169023922264211005098194, 5.54625134389718110930114170439, 5.71768062725796861053797817553, 5.94052238633313809771556097036
Plot not available for L-functions of degree greater than 10.
