| L(s) = 1 | − 5·3-s + 9·4-s − 4·7-s + 9·9-s − 45·12-s + 8·13-s + 37·16-s + 36·19-s + 20·21-s + 93·25-s − 10·27-s − 36·28-s + 46·31-s + 81·36-s − 90·37-s − 40·39-s − 96·43-s − 185·48-s − 120·49-s + 72·52-s − 180·57-s + 24·61-s − 36·63-s + 81·64-s − 58·67-s + 284·73-s − 465·75-s + ⋯ |
| L(s) = 1 | − 5/3·3-s + 9/4·4-s − 4/7·7-s + 9-s − 3.75·12-s + 8/13·13-s + 2.31·16-s + 1.89·19-s + 0.952·21-s + 3.71·25-s − 0.370·27-s − 9/7·28-s + 1.48·31-s + 9/4·36-s − 2.43·37-s − 1.02·39-s − 2.23·43-s − 3.85·48-s − 2.44·49-s + 1.38·52-s − 3.15·57-s + 0.393·61-s − 4/7·63-s + 1.26·64-s − 0.865·67-s + 3.89·73-s − 6.19·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.591532393\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.591532393\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2^2$ | \( 1 + 5 T + 16 T^{2} + 5 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 9 T^{2} + 11 p^{2} T^{4} - 9 p^{4} T^{6} + p^{8} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 - 93 T^{2} + 3404 T^{4} - 93 p^{4} T^{6} + p^{8} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + 2 T + 66 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 - 4 T + 210 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 216 T^{2} + 149006 T^{4} + 216 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 18 T + 506 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 477 T^{2} + 607580 T^{4} - 477 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 188 T^{2} + 246 p^{2} T^{4} + 188 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 23 T + 1848 T^{2} - 23 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 45 T + 2840 T^{2} + 45 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 5460 T^{2} + 13000934 T^{4} - 5460 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 48 T + 4142 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 6120 T^{2} + 18979214 T^{4} - 6120 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 3112 T^{2} + 2490798 T^{4} - 3112 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 9921 T^{2} + 45659744 T^{4} - 9921 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 12 T + 4178 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 29 T + 6804 T^{2} + 29 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 19933 T^{2} + 150145500 T^{4} - 19933 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 142 T + 15402 T^{2} - 142 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 38 T + 7266 T^{2} - 38 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 22120 T^{2} + 214834542 T^{4} - 22120 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 10897 T^{2} + 51258576 T^{4} - 10897 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 109 T + 132 p T^{2} - 109 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.080784392590254856110178216534, −7.898725675774542855132189876123, −7.28742796022470551230842081172, −7.05943048157780258827524825755, −6.91415010881811839049358458556, −6.58937285971004447660358179237, −6.55866897057035480557951979891, −6.44151999897873301883755777738, −6.38752676768062143798355637348, −5.64658660148817319180914448752, −5.43876514375927257514435125039, −5.24573634063919200640776344882, −5.17980139884944245972525199799, −4.77089283749589369907728052037, −4.56295122647449617063353668300, −3.88620190720541997643853030273, −3.37727888797820370120226749668, −3.33654764919392249792525378454, −3.08011797297600168016539549896, −2.68014588804427720236458805973, −2.34501074048050583273722498603, −1.72738474262433880498118009973, −1.30165593428722335016829528314, −1.10750588875625658280892590154, −0.39065716315635361243385313878,
0.39065716315635361243385313878, 1.10750588875625658280892590154, 1.30165593428722335016829528314, 1.72738474262433880498118009973, 2.34501074048050583273722498603, 2.68014588804427720236458805973, 3.08011797297600168016539549896, 3.33654764919392249792525378454, 3.37727888797820370120226749668, 3.88620190720541997643853030273, 4.56295122647449617063353668300, 4.77089283749589369907728052037, 5.17980139884944245972525199799, 5.24573634063919200640776344882, 5.43876514375927257514435125039, 5.64658660148817319180914448752, 6.38752676768062143798355637348, 6.44151999897873301883755777738, 6.55866897057035480557951979891, 6.58937285971004447660358179237, 6.91415010881811839049358458556, 7.05943048157780258827524825755, 7.28742796022470551230842081172, 7.898725675774542855132189876123, 8.080784392590254856110178216534
