| L(s) = 1 | − 3-s + 7-s − 2·13-s + 19-s − 21-s − 25-s + 31-s + 37-s + 2·39-s + 8·43-s + 49-s − 57-s + 61-s − 4·67-s + 73-s + 75-s + 79-s − 2·91-s − 93-s + 97-s + 103-s − 4·109-s − 111-s + 127-s − 8·129-s + 131-s + 133-s + ⋯ |
| L(s) = 1 | − 3-s + 7-s − 2·13-s + 19-s − 21-s − 25-s + 31-s + 37-s + 2·39-s + 8·43-s + 49-s − 57-s + 61-s − 4·67-s + 73-s + 75-s + 79-s − 2·91-s − 93-s + 97-s + 103-s − 4·109-s − 111-s + 127-s − 8·129-s + 131-s + 133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7913505820\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7913505820\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | | \( 1 \) |
| good | 5 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 7 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 13 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 17 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 19 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 29 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 31 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 37 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 41 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 43 | $C_1$ | \( ( 1 - T )^{8} \) |
| 47 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 53 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 59 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 61 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 71 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 73 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 83 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
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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
−7.19170022028651717731064058856, −6.77091505978560696282802040059, −6.32386884006124960386493964478, −6.30716183709795566926610724160, −6.20357058340496989670000213472, −5.74941391031333380635649507206, −5.72239156696724876445176673654, −5.46592999778632449372970400742, −5.40072237263758534397085337937, −5.02224790027657226262992655050, −4.87766048960526334035773386020, −4.53281179351557271959370987158, −4.24481931812450309555751200454, −4.22582064560975869019590040608, −4.03137384695765897315210433337, −3.82336048142633007949087665322, −3.14207717079243753562847926677, −2.88948847286368683400304257951, −2.83614822057332204978132633076, −2.28336631182037581080680644671, −2.24444897088159940053616511441, −2.18344515155380079915664516763, −1.25928467382704474861453913136, −1.13345157799083553098973270012, −0.69952496655136278319894867673,
0.69952496655136278319894867673, 1.13345157799083553098973270012, 1.25928467382704474861453913136, 2.18344515155380079915664516763, 2.24444897088159940053616511441, 2.28336631182037581080680644671, 2.83614822057332204978132633076, 2.88948847286368683400304257951, 3.14207717079243753562847926677, 3.82336048142633007949087665322, 4.03137384695765897315210433337, 4.22582064560975869019590040608, 4.24481931812450309555751200454, 4.53281179351557271959370987158, 4.87766048960526334035773386020, 5.02224790027657226262992655050, 5.40072237263758534397085337937, 5.46592999778632449372970400742, 5.72239156696724876445176673654, 5.74941391031333380635649507206, 6.20357058340496989670000213472, 6.30716183709795566926610724160, 6.32386884006124960386493964478, 6.77091505978560696282802040059, 7.19170022028651717731064058856
