| L(s) = 1 | + 8·7-s − 32·13-s − 32·19-s + 8·25-s + 8·31-s − 104·37-s − 112·43-s − 156·49-s − 120·61-s − 336·67-s − 112·73-s + 152·79-s − 256·91-s − 352·97-s + 344·103-s − 512·109-s − 12·121-s + 127-s + 131-s − 256·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 8/7·7-s − 2.46·13-s − 1.68·19-s + 8/25·25-s + 8/31·31-s − 2.81·37-s − 2.60·43-s − 3.18·49-s − 1.96·61-s − 5.01·67-s − 1.53·73-s + 1.92·79-s − 2.81·91-s − 3.62·97-s + 3.33·103-s − 4.69·109-s − 0.0991·121-s + 0.00787·127-s + 0.00763·131-s − 1.92·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.001298057981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.001298057981\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 914 T^{4} - 8 p^{4} T^{6} + p^{8} T^{8} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{4} \) |
| 11 | $D_4\times C_2$ | \( 1 + 12 T^{2} - 21370 T^{4} + 12 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 16 T + 314 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 800 T^{2} + 325634 T^{4} - 800 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 16 T + 434 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 564 T^{2} + 436454 T^{4} - 564 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2600 T^{2} + 3045074 T^{4} - 2600 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 4 T + 518 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 26 T + p^{2} T^{2} )^{4} \) |
| 41 | $D_4\times C_2$ | \( 1 - 5888 T^{2} + 14148290 T^{4} - 5888 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 56 T + 3074 T^{2} + 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 1652 T^{2} + 2309030 T^{4} - 1652 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 8680 T^{2} + 33219474 T^{4} - 8680 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 1554 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 60 T + 6934 T^{2} + 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 168 T + 15682 T^{2} + 168 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 12596 T^{2} + 79584614 T^{4} - 12596 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 56 T - 1230 T^{2} + 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 76 T + 12518 T^{2} - 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 24244 T^{2} + 241403334 T^{4} - 24244 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 10176 T^{2} + 39402434 T^{4} - 10176 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 176 T + 26210 T^{2} + 176 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−6.79555063446752741446662256267, −6.73713820310023488101269777040, −6.54552683994311667345799058753, −5.95441612045793195681116869320, −5.93620468153749277454413876162, −5.86288128257062920556713084279, −5.16998773775743773134999822699, −5.13199619853342373327668527149, −5.00710660589464606124336848130, −4.72380727534061633431401841925, −4.65629621458645987295734857007, −4.43405349313545469619952358687, −4.16431898463927211778232036373, −3.67419893584542090595933142029, −3.52239866313546453349953498752, −3.17960780901853170246344158875, −2.82022888404971945574292373246, −2.77465300297593931872001978068, −2.43777562272372934685779554592, −1.88064110833168321677345786488, −1.63236088129883495789062116671, −1.55665151803461014510721844979, −1.46465389196128520153040320226, −0.32949543858586893460626883911, −0.009127143153893421929138242284,
0.009127143153893421929138242284, 0.32949543858586893460626883911, 1.46465389196128520153040320226, 1.55665151803461014510721844979, 1.63236088129883495789062116671, 1.88064110833168321677345786488, 2.43777562272372934685779554592, 2.77465300297593931872001978068, 2.82022888404971945574292373246, 3.17960780901853170246344158875, 3.52239866313546453349953498752, 3.67419893584542090595933142029, 4.16431898463927211778232036373, 4.43405349313545469619952358687, 4.65629621458645987295734857007, 4.72380727534061633431401841925, 5.00710660589464606124336848130, 5.13199619853342373327668527149, 5.16998773775743773134999822699, 5.86288128257062920556713084279, 5.93620468153749277454413876162, 5.95441612045793195681116869320, 6.54552683994311667345799058753, 6.73713820310023488101269777040, 6.79555063446752741446662256267
