| L(s) = 1 | − 4·3-s + 4·9-s − 4·13-s − 8·25-s − 4·27-s − 12·29-s − 4·31-s + 16·39-s + 16·41-s − 28·47-s − 20·49-s − 36·59-s − 32·71-s + 20·73-s + 32·75-s + 32·81-s + 48·87-s + 16·93-s − 32·101-s − 16·117-s − 34·121-s − 64·123-s + 127-s + 131-s + 137-s + 139-s + 112·141-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 4/3·9-s − 1.10·13-s − 8/5·25-s − 0.769·27-s − 2.22·29-s − 0.718·31-s + 2.56·39-s + 2.49·41-s − 4.08·47-s − 2.85·49-s − 4.68·59-s − 3.79·71-s + 2.34·73-s + 3.69·75-s + 32/9·81-s + 5.14·87-s + 1.65·93-s − 3.18·101-s − 1.47·117-s − 3.09·121-s − 5.77·123-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 9.43·141-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 23^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 23^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( ( 1 + 2 T + 4 T^{2} + 10 T^{3} + 4 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 5 | \( 1 + 8 T^{2} + 7 p T^{4} + 72 T^{6} + 7 p^{3} T^{8} + 8 p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 + 20 T^{2} + 29 p T^{4} + 1536 T^{6} + 29 p^{3} T^{8} + 20 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 + 34 T^{2} + 439 T^{4} + 4084 T^{6} + 439 p^{2} T^{8} + 34 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( ( 1 + 2 T + 34 T^{2} + 50 T^{3} + 34 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 17 | \( 1 + 86 T^{2} + 3311 T^{4} + 72684 T^{6} + 3311 p^{2} T^{8} + 86 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( 1 + 28 T^{2} + 939 T^{4} + 18232 T^{6} + 939 p^{2} T^{8} + 28 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 + 6 T + 38 T^{2} + 266 T^{3} + 38 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 + 2 T + 2 p T^{2} + 156 T^{3} + 2 p^{2} T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 + 144 T^{2} + 10691 T^{4} + 489976 T^{6} + 10691 p^{2} T^{8} + 144 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 - 8 T + 94 T^{2} - 672 T^{3} + 94 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 + 96 T^{2} + 7391 T^{4} + 331984 T^{6} + 7391 p^{2} T^{8} + 96 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( ( 1 + 14 T + 194 T^{2} + 1372 T^{3} + 194 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 53 | \( 1 + 250 T^{2} + 28363 T^{4} + 1899604 T^{6} + 28363 p^{2} T^{8} + 250 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 18 T + 269 T^{2} + 2236 T^{3} + 269 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( 1 + 114 T^{2} + 10011 T^{4} + 627204 T^{6} + 10011 p^{2} T^{8} + 114 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( 1 + 210 T^{2} + 25095 T^{4} + 1997268 T^{6} + 25095 p^{2} T^{8} + 210 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 16 T + 292 T^{2} + 2388 T^{3} + 292 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( ( 1 - 10 T + 130 T^{2} - 574 T^{3} + 130 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( 1 + 254 T^{2} + 35939 T^{4} + 3450204 T^{6} + 35939 p^{2} T^{8} + 254 p^{4} T^{10} + p^{6} T^{12} \) |
| 83 | \( 1 + 130 T^{2} + 19303 T^{4} + 1525684 T^{6} + 19303 p^{2} T^{8} + 130 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 + 300 T^{2} + 49767 T^{4} + 5422552 T^{6} + 49767 p^{2} T^{8} + 300 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 + 238 T^{2} + 46143 T^{4} + 4893916 T^{6} + 46143 p^{2} T^{8} + 238 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−4.27590974204743816389876990615, −4.24531838588701296659755852512, −4.08917808204769090995446962315, −3.88282661193919139138619754436, −3.86972633047625189316315381328, −3.74395668093524686533114612996, −3.73678049725916512244005464647, −3.42408212364598233326915530406, −3.13619083383609369655240084517, −3.11234132142746327167318731373, −3.09500800400068841183558911339, −2.96781990905129518685917235701, −2.87542867560695384572395069849, −2.42440787150751255667425212205, −2.28817295526490270063555609478, −2.26779021348335387540871704239, −2.25286023743205299451249203535, −2.01607145751329164125121781944, −1.72061395557681270920678352920, −1.53013802084351492203895970284, −1.46261269449941461145726374447, −1.37706944205340830270360669107, −1.18289872844398677629115436416, −1.12470323917585585785559974122, −0.63534836411758074573355604141, 0, 0, 0, 0, 0, 0,
0.63534836411758074573355604141, 1.12470323917585585785559974122, 1.18289872844398677629115436416, 1.37706944205340830270360669107, 1.46261269449941461145726374447, 1.53013802084351492203895970284, 1.72061395557681270920678352920, 2.01607145751329164125121781944, 2.25286023743205299451249203535, 2.26779021348335387540871704239, 2.28817295526490270063555609478, 2.42440787150751255667425212205, 2.87542867560695384572395069849, 2.96781990905129518685917235701, 3.09500800400068841183558911339, 3.11234132142746327167318731373, 3.13619083383609369655240084517, 3.42408212364598233326915530406, 3.73678049725916512244005464647, 3.74395668093524686533114612996, 3.86972633047625189316315381328, 3.88282661193919139138619754436, 4.08917808204769090995446962315, 4.24531838588701296659755852512, 4.27590974204743816389876990615
Plot not available for L-functions of degree greater than 10.
