| L(s) = 1 | − 6·2-s + 18·4-s + 10·7-s − 36·8-s + 9-s + 2·11-s − 60·14-s + 52·16-s + 6·17-s − 6·18-s + 2·19-s − 12·22-s + 6·23-s + 180·28-s − 4·29-s − 6·31-s − 48·32-s − 36·34-s + 18·36-s − 18·37-s − 12·38-s + 4·41-s + 36·44-s − 36·46-s + 61·49-s + 36·53-s − 360·56-s + ⋯ |
| L(s) = 1 | − 4.24·2-s + 9·4-s + 3.77·7-s − 12.7·8-s + 1/3·9-s + 0.603·11-s − 16.0·14-s + 13·16-s + 1.45·17-s − 1.41·18-s + 0.458·19-s − 2.55·22-s + 1.25·23-s + 34.0·28-s − 0.742·29-s − 1.07·31-s − 8.48·32-s − 6.17·34-s + 3·36-s − 2.95·37-s − 1.94·38-s + 0.624·41-s + 5.42·44-s − 5.30·46-s + 61/7·49-s + 4.94·53-s − 48.1·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6392221296\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6392221296\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| good | 2 | $C_2$$\times$$C_2^2$ | \( ( 1 + p T + p T^{2} )^{2}( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} ) \) |
| 11 | $D_4\times C_2$ | \( 1 - 2 T - 16 T^{2} + 4 T^{3} + 235 T^{4} + 4 p T^{5} - 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 14 T^{2} + 15 p T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 6 T + 24 T^{2} - 72 T^{3} + 59 T^{4} - 72 p T^{5} + 24 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 2 T - 23 T^{2} + 22 T^{3} + 292 T^{4} + 22 p T^{5} - 23 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 6 T + 52 T^{2} - 240 T^{3} + 1347 T^{4} - 240 p T^{5} + 52 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 2 T + 32 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 6 T - 23 T^{2} - 18 T^{3} + 1404 T^{4} - 18 p T^{5} - 23 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 18 T + 205 T^{2} + 1746 T^{3} + 12036 T^{4} + 1746 p T^{5} + 205 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 2 T + 80 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 110 T^{2} + 6291 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 90 T^{2} + 5891 T^{4} + 90 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 36 T + 642 T^{2} - 7560 T^{3} + 64187 T^{4} - 7560 p T^{5} + 642 p^{2} T^{6} - 36 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 10 T - 16 T^{2} + 20 T^{3} + 4075 T^{4} + 20 p T^{5} - 16 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 30 T + 473 T^{2} - 5190 T^{3} + 45540 T^{4} - 5190 p T^{5} + 473 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 2 T + 116 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 30 T + 505 T^{2} - 6150 T^{3} + 58596 T^{4} - 6150 p T^{5} + 505 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 6 T - 23 T^{2} + 594 T^{3} - 5604 T^{4} + 594 p T^{5} - 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 8586 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 6 T - 4 T^{2} + 828 T^{3} - 9525 T^{4} + 828 p T^{5} - 4 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 13254 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.902901532447817267410596539800, −7.71456437367338434598713672274, −7.48052191516048858585644476890, −7.45910403246871476877594419935, −7.39999107185373102339085595692, −6.81795350565640618371458417213, −6.69952447536074903986805879981, −6.64899230199804599491697035486, −5.81052105334269405746371238762, −5.66465800535425033931959532244, −5.43682811367613161335282733701, −5.06017043934034719462328163870, −4.94380827983789894853145636433, −4.88605919926622449298858228661, −4.22608444829518348100613054512, −3.75150188242948291701146128565, −3.66807239622448054706647150160, −3.43695980300812316507625698378, −2.26998119543594401573493276492, −2.24208178343154337055165461172, −2.17877235028455554693608118324, −1.72609792241351778106834425704, −0.975275579198860834757675919719, −0.952338313780787739264269641214, −0.937870288855792845004158091441,
0.937870288855792845004158091441, 0.952338313780787739264269641214, 0.975275579198860834757675919719, 1.72609792241351778106834425704, 2.17877235028455554693608118324, 2.24208178343154337055165461172, 2.26998119543594401573493276492, 3.43695980300812316507625698378, 3.66807239622448054706647150160, 3.75150188242948291701146128565, 4.22608444829518348100613054512, 4.88605919926622449298858228661, 4.94380827983789894853145636433, 5.06017043934034719462328163870, 5.43682811367613161335282733701, 5.66465800535425033931959532244, 5.81052105334269405746371238762, 6.64899230199804599491697035486, 6.69952447536074903986805879981, 6.81795350565640618371458417213, 7.39999107185373102339085595692, 7.45910403246871476877594419935, 7.48052191516048858585644476890, 7.71456437367338434598713672274, 7.902901532447817267410596539800
