| L(s) = 1 | − 2·2-s + 4-s + 4·5-s + 2·7-s + 2·8-s − 8·10-s + 2·11-s + 6·13-s − 4·14-s − 4·16-s − 28·17-s − 4·19-s + 4·20-s − 4·22-s + 2·23-s + 11·25-s − 12·26-s + 2·28-s + 10·29-s − 6·31-s + 2·32-s + 56·34-s + 8·35-s + 8·37-s + 8·38-s + 8·40-s + 12·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s + 1.78·5-s + 0.755·7-s + 0.707·8-s − 2.52·10-s + 0.603·11-s + 1.66·13-s − 1.06·14-s − 16-s − 6.79·17-s − 0.917·19-s + 0.894·20-s − 0.852·22-s + 0.417·23-s + 11/5·25-s − 2.35·26-s + 0.377·28-s + 1.85·29-s − 1.07·31-s + 0.353·32-s + 9.60·34-s + 1.35·35-s + 1.31·37-s + 1.29·38-s + 1.26·40-s + 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.992735316\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.992735316\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| good | 5 | $C_2$$\times$$C_2^2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 11 | $D_4\times C_2$ | \( 1 - 2 T + 8 T^{2} + 52 T^{3} - 149 T^{4} + 52 p T^{5} + 8 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2$$\times$$C_2^2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 17 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 19 | $D_{4}$ | \( ( 1 + 2 T + 36 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 2 T - 16 T^{2} + 52 T^{3} - 221 T^{4} + 52 p T^{5} - 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + 71 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \) |
| 31 | $D_4\times C_2$ | \( 1 + 6 T - 8 T^{2} - 108 T^{3} - 141 T^{4} - 108 p T^{5} - 8 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 12 T + 38 T^{2} - 288 T^{3} + 3651 T^{4} - 288 p T^{5} + 38 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 4 T - 62 T^{2} - 32 T^{3} + 3547 T^{4} - 32 p T^{5} - 62 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 6 T - 64 T^{2} - 36 T^{3} + 6099 T^{4} - 36 p T^{5} - 64 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 2 T - 88 T^{2} - 52 T^{3} + 4747 T^{4} - 52 p T^{5} - 88 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 6 T - 47 T^{2} + 234 T^{3} + 972 T^{4} + 234 p T^{5} - 47 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 10 T - 56 T^{2} - 220 T^{3} + 13147 T^{4} - 220 p T^{5} - 56 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 20 T + 230 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 20 T + 243 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 6 T + 16 T^{2} - 828 T^{3} - 8685 T^{4} - 828 p T^{5} + 16 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 2 T + 80 T^{2} + 484 T^{3} - 1445 T^{4} + 484 p T^{5} + 80 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 6 T + 139 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 8 T - 98 T^{2} - 256 T^{3} + 10627 T^{4} - 256 p T^{5} - 98 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.92448964616507388267743805429, −6.80506775421695516142253494543, −6.62858979995626503448401987891, −6.37099562595878266180189287326, −6.31089670180172799703222396171, −6.05988477447521291914017864234, −5.85967622638866360560991853017, −5.46795525338221720890159210045, −5.22599904277464858287026612790, −5.02029252943026081262982644510, −4.50254309061703550009047879704, −4.42175339431111310602425777761, −4.29312548658099977542858581654, −4.27597932065107134121208381388, −3.93823311093402333688561153616, −3.56303409935884354290170176006, −2.77467198021771906339500641098, −2.75358899345313692942159518059, −2.33026107097674820849612917685, −2.24851710181509969617105862254, −1.90393965717513917812347915940, −1.81852840755495423695993278953, −1.26416446183049235854952924082, −0.78077031696565700672146600386, −0.48399585579180060737933102351,
0.48399585579180060737933102351, 0.78077031696565700672146600386, 1.26416446183049235854952924082, 1.81852840755495423695993278953, 1.90393965717513917812347915940, 2.24851710181509969617105862254, 2.33026107097674820849612917685, 2.75358899345313692942159518059, 2.77467198021771906339500641098, 3.56303409935884354290170176006, 3.93823311093402333688561153616, 4.27597932065107134121208381388, 4.29312548658099977542858581654, 4.42175339431111310602425777761, 4.50254309061703550009047879704, 5.02029252943026081262982644510, 5.22599904277464858287026612790, 5.46795525338221720890159210045, 5.85967622638866360560991853017, 6.05988477447521291914017864234, 6.31089670180172799703222396171, 6.37099562595878266180189287326, 6.62858979995626503448401987891, 6.80506775421695516142253494543, 6.92448964616507388267743805429
