| L(s) = 1 | − 4·2-s − 2·3-s + 10·4-s + 3·5-s + 8·6-s − 20·8-s − 3·9-s − 12·10-s − 3·11-s − 20·12-s + 4·13-s − 6·15-s + 35·16-s − 3·17-s + 12·18-s + 10·19-s + 30·20-s + 12·22-s + 9·23-s + 40·24-s + 4·25-s − 16·26-s + 14·27-s + 6·29-s + 24·30-s − 8·31-s − 56·32-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 1.15·3-s + 5·4-s + 1.34·5-s + 3.26·6-s − 7.07·8-s − 9-s − 3.79·10-s − 0.904·11-s − 5.77·12-s + 1.10·13-s − 1.54·15-s + 35/4·16-s − 0.727·17-s + 2.82·18-s + 2.29·19-s + 6.70·20-s + 2.55·22-s + 1.87·23-s + 8.16·24-s + 4/5·25-s − 3.13·26-s + 2.69·27-s + 1.11·29-s + 4.38·30-s − 1.43·31-s − 9.89·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6665274214\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6665274214\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + T )^{4} \) |
| 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$$\times$$C_2^2$ | \( ( 1 - 3 T + p T^{2} )^{2}( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} ) \) |
| 11 | $D_4\times C_2$ | \( 1 + 3 T - 7 T^{2} - 18 T^{3} + 36 T^{4} - 18 p T^{5} - 7 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 3 T - 19 T^{2} - 18 T^{3} + 342 T^{4} - 18 p T^{5} - 19 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 - 9 T + p T^{2} )^{2}( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} ) \) |
| 29 | $D_4\times C_2$ | \( 1 - 6 T + 2 T^{2} + 144 T^{3} - 729 T^{4} + 144 p T^{5} + 2 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 15 T + 95 T^{2} - 720 T^{3} + 5994 T^{4} - 720 p T^{5} + 95 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + T - 11 T^{2} - 74 T^{3} - 1748 T^{4} - 74 p T^{5} - 11 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $D_4\times C_2$ | \( 1 - 6 T - 46 T^{2} + 144 T^{3} + 2007 T^{4} + 144 p T^{5} - 46 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 3 T + 46 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 11 T + 78 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 3 T + 70 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 7 T - 35 T^{2} + 434 T^{3} - 1850 T^{4} + 434 p T^{5} - 35 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 7 T + 96 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 12 T + 74 T^{2} + 1152 T^{3} - 13941 T^{4} + 1152 p T^{5} + 74 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 18 T + 98 T^{2} - 864 T^{3} + 14319 T^{4} - 864 p T^{5} + 98 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - T - 119 T^{2} + 74 T^{3} + 4894 T^{4} + 74 p T^{5} - 119 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.34189556058844969738082064704, −6.96005265771186889141204050122, −6.81442951393795254157027431772, −6.77312937440100380232909430235, −6.38179615249510903771827153538, −6.27145284296263410050031464904, −6.11711128522442438731097350319, −5.64755148465717659849735431684, −5.46235514610557300153449040537, −5.35723657218435868243043035227, −5.24090715196535114281999016393, −4.91540282621348028589242064689, −4.84284062495842582755710351319, −3.90315696017910636251021454334, −3.66637222612032362078387306019, −3.53199008311516285914807976365, −3.21048608694487194339911942489, −2.55097256537709003880837194698, −2.50244670026914170383482709836, −2.43975375425703291348909996971, −2.16626085689598899485953502931, −1.29597139505715774102254452748, −1.14175170795472192870097505512, −0.851760292146256159865905463597, −0.49726250061093211750543280831,
0.49726250061093211750543280831, 0.851760292146256159865905463597, 1.14175170795472192870097505512, 1.29597139505715774102254452748, 2.16626085689598899485953502931, 2.43975375425703291348909996971, 2.50244670026914170383482709836, 2.55097256537709003880837194698, 3.21048608694487194339911942489, 3.53199008311516285914807976365, 3.66637222612032362078387306019, 3.90315696017910636251021454334, 4.84284062495842582755710351319, 4.91540282621348028589242064689, 5.24090715196535114281999016393, 5.35723657218435868243043035227, 5.46235514610557300153449040537, 5.64755148465717659849735431684, 6.11711128522442438731097350319, 6.27145284296263410050031464904, 6.38179615249510903771827153538, 6.77312937440100380232909430235, 6.81442951393795254157027431772, 6.96005265771186889141204050122, 7.34189556058844969738082064704
