| L(s) = 1 | − 2-s − 2·3-s − 2·4-s + 2·6-s + 2·8-s − 9-s − 8·11-s + 4·12-s − 8·13-s + 16-s − 2·17-s + 18-s − 3·19-s + 8·22-s + 4·23-s − 4·24-s + 8·26-s + 4·27-s − 10·29-s + 4·31-s + 32-s + 16·33-s + 2·34-s + 2·36-s − 20·37-s + 3·38-s + 16·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 4-s + 0.816·6-s + 0.707·8-s − 1/3·9-s − 2.41·11-s + 1.15·12-s − 2.21·13-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.688·19-s + 1.70·22-s + 0.834·23-s − 0.816·24-s + 1.56·26-s + 0.769·27-s − 1.85·29-s + 0.718·31-s + 0.176·32-s + 2.78·33-s + 0.342·34-s + 1/3·36-s − 3.28·37-s + 0.486·38-s + 2.56·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T + 3 T^{2} + 3 T^{3} + 3 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 3 | $S_4\times C_2$ | \( 1 + 2 T + 5 T^{2} + 8 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 5 T^{2} - 16 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 8 T + 41 T^{2} + 160 T^{3} + 41 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 8 T + 51 T^{2} + 212 T^{3} + 51 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 2 T + 15 T^{2} - 36 T^{3} + 15 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 4 T + 61 T^{2} - 168 T^{3} + 61 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 10 T + 99 T^{2} + 540 T^{3} + 99 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 4 T + 45 T^{2} - 184 T^{3} + 45 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 20 T + 235 T^{2} + 1724 T^{3} + 235 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 2 T + 87 T^{2} + 60 T^{3} + 87 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 4 T - 15 T^{2} + 248 T^{3} - 15 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 125 T^{2} - 16 T^{3} + 125 p T^{4} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 16 T + 235 T^{2} + 1788 T^{3} + 235 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 20 T + 289 T^{2} + 2520 T^{3} + 289 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 2 T + 99 T^{2} + 476 T^{3} + 99 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 2 T + 125 T^{2} + 384 T^{3} + 125 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 4 T + 133 T^{2} + 504 T^{3} + 133 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 2 T + 199 T^{2} + 284 T^{3} + 199 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 45 T^{2} - 160 T^{3} + 45 p T^{4} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 32 T + 577 T^{2} - 6384 T^{3} + 577 p T^{4} - 32 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 2 T + 135 T^{2} + 324 T^{3} + 135 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 20 T + 231 T^{2} + 2132 T^{3} + 231 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.31803516712566130100771050541, −9.910143149772269824359035553464, −9.596030474136190821664917785490, −9.538426248023707266528190880262, −9.066418741591584822201264381691, −8.726054419556630194650702462318, −8.693550340092958874569697763055, −8.028091385468656552884780165313, −7.87462289202091224606469301332, −7.52786815767506675418148855799, −7.48024771249110438457370552182, −6.81730870589100847445965028483, −6.53381958851570359278476920409, −6.24145341271270623921407924606, −5.61833942145190438759590697906, −5.41800378276575073717815413676, −5.03625291677403996088410488284, −4.99171169998799150443949090789, −4.73287086874634028716202162561, −4.21324210655885383639464087734, −3.58462042181120631379624837386, −2.96067152540332391416046639408, −2.81151268124156327008609849566, −2.18381579025341456127530937661, −1.64756415375780415218626594701, 0, 0, 0,
1.64756415375780415218626594701, 2.18381579025341456127530937661, 2.81151268124156327008609849566, 2.96067152540332391416046639408, 3.58462042181120631379624837386, 4.21324210655885383639464087734, 4.73287086874634028716202162561, 4.99171169998799150443949090789, 5.03625291677403996088410488284, 5.41800378276575073717815413676, 5.61833942145190438759590697906, 6.24145341271270623921407924606, 6.53381958851570359278476920409, 6.81730870589100847445965028483, 7.48024771249110438457370552182, 7.52786815767506675418148855799, 7.87462289202091224606469301332, 8.028091385468656552884780165313, 8.693550340092958874569697763055, 8.726054419556630194650702462318, 9.066418741591584822201264381691, 9.538426248023707266528190880262, 9.596030474136190821664917785490, 9.910143149772269824359035553464, 10.31803516712566130100771050541
