| L(s) = 1 | − 20·7-s − 68·13-s − 17·16-s + 24·19-s + 60·25-s − 56·31-s + 72·37-s − 12·43-s + 84·49-s − 24·61-s − 52·67-s − 128·73-s − 136·79-s + 1.36e3·91-s + 176·97-s − 148·103-s − 20·109-s + 340·112-s + 127-s + 131-s − 480·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 2.85·7-s − 5.23·13-s − 1.06·16-s + 1.26·19-s + 12/5·25-s − 1.80·31-s + 1.94·37-s − 0.279·43-s + 12/7·49-s − 0.393·61-s − 0.776·67-s − 1.75·73-s − 1.72·79-s + 14.9·91-s + 1.81·97-s − 1.43·103-s − 0.183·109-s + 3.03·112-s + 0.00787·127-s + 0.00763·131-s − 3.60·133-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3890264209\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3890264209\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 + 17 T^{4} + p^{8} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 - 12 p T^{2} + 71 p^{2} T^{4} - 12 p^{5} T^{6} + p^{8} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + 10 T + 108 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 + 34 T + 567 T^{2} + 34 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 996 T^{2} + 411671 T^{4} - 996 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 12 T + 218 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1320 T^{2} + 857042 T^{4} - 1320 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1324 T^{2} + 1822431 T^{4} - 1324 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 28 T + 618 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 36 T + 2687 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 1500 T^{2} + 3920807 T^{4} + 1500 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 6 T - 628 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 8196 T^{2} + 26498966 T^{4} - 8196 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 10060 T^{2} + 40757727 T^{4} - 10060 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 2160 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 12 T + 6518 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 26 T + 9132 T^{2} + 26 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 13120 T^{2} + 88016322 T^{4} - 13120 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 64 T + 11142 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 68 T + 12678 T^{2} + 68 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 11140 T^{2} + 86575542 T^{4} - 11140 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 8260 T^{2} + 142190247 T^{4} - 8260 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 88 T + 17379 T^{2} - 88 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−6.82253543303562576483451262313, −6.72081226804023743488054492596, −6.70763369001322924531814379214, −6.26323388698159434175822723369, −5.85673269928520070977571078377, −5.74148302220411552187854893645, −5.56346588750448155873263563187, −5.25007003259947130243381452932, −5.00770489036060016399701680442, −4.69886566330150974334129433235, −4.56313406822614476917612312214, −4.37719813385210866804740822914, −4.36208381054812495490863213357, −3.51809779073078276316742084605, −3.48121381102572871161516502510, −2.98042363271584453514367179457, −2.95727290582062952216482524724, −2.87541305349938153512127034466, −2.60939685567580961909883463120, −2.23345313990705584542537128422, −1.87946633957535477012872002300, −1.59177747811205024505149652427, −0.70198565184991028268056934745, −0.46652809764785180634861064418, −0.18223111205969090886899805041,
0.18223111205969090886899805041, 0.46652809764785180634861064418, 0.70198565184991028268056934745, 1.59177747811205024505149652427, 1.87946633957535477012872002300, 2.23345313990705584542537128422, 2.60939685567580961909883463120, 2.87541305349938153512127034466, 2.95727290582062952216482524724, 2.98042363271584453514367179457, 3.48121381102572871161516502510, 3.51809779073078276316742084605, 4.36208381054812495490863213357, 4.37719813385210866804740822914, 4.56313406822614476917612312214, 4.69886566330150974334129433235, 5.00770489036060016399701680442, 5.25007003259947130243381452932, 5.56346588750448155873263563187, 5.74148302220411552187854893645, 5.85673269928520070977571078377, 6.26323388698159434175822723369, 6.70763369001322924531814379214, 6.72081226804023743488054492596, 6.82253543303562576483451262313
