| L(s) = 1 | + 2·2-s − 3-s + 4-s + 6·5-s − 2·6-s − 2·8-s + 3·9-s + 12·10-s + 6·11-s − 12-s − 4·13-s − 6·15-s − 4·16-s + 3·17-s + 6·18-s − 10·19-s + 6·20-s + 12·22-s − 18·23-s + 2·24-s + 19·25-s − 8·26-s − 8·27-s + 6·29-s − 12·30-s − 4·31-s − 2·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 1/2·4-s + 2.68·5-s − 0.816·6-s − 0.707·8-s + 9-s + 3.79·10-s + 1.80·11-s − 0.288·12-s − 1.10·13-s − 1.54·15-s − 16-s + 0.727·17-s + 1.41·18-s − 2.29·19-s + 1.34·20-s + 2.55·22-s − 3.75·23-s + 0.408·24-s + 19/5·25-s − 1.56·26-s − 1.53·27-s + 1.11·29-s − 2.19·30-s − 0.718·31-s − 0.353·32-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
| \(L(1)\) |
\(\approx\) |
\(3.218860377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.218860377\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 - 3 T + 16 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( ( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 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 | $C_2^2$ | \( ( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 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 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 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 | $D_4\times C_2$ | \( 1 - 3 T - 37 T^{2} + 216 T^{3} - 1896 T^{4} + 216 p T^{5} - 37 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 11 T + 43 T^{2} + 484 T^{3} - 5018 T^{4} + 484 p T^{5} + 43 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + 13 T + p T^{2} )^{2}( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} ) \) |
| 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 | $D_4\times C_2$ | \( 1 + 7 T - 47 T^{2} - 434 T^{3} - 896 T^{4} - 434 p T^{5} - 47 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 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
−6.97362593833939523540628652736, −6.89817361803044741509482742612, −6.79131042432532082199724315142, −6.40302877460966493467995695972, −6.24708437463745515654714362060, −6.12706998132374100489403724241, −5.79079032941488174358061833422, −5.58281611318310938060718697071, −5.53418972260422996179318487716, −5.28861207707756843996798095166, −5.11487609696476299226968965753, −4.53408477818591859292982505535, −4.27488739828705691199349531519, −4.23745593571510031171132037109, −4.22904794972117482123567903988, −3.72321739260956098462027395558, −3.58691966846064477060217541732, −2.93956834741724938854186651480, −2.88134115319943188416323233101, −2.27914554741916594442797353271, −2.04877909036969525082440318885, −1.86701289254042232053151519910, −1.60944633102318667180547365507, −1.36223859627416445933931523346, −0.29680947834455573733651518480,
0.29680947834455573733651518480, 1.36223859627416445933931523346, 1.60944633102318667180547365507, 1.86701289254042232053151519910, 2.04877909036969525082440318885, 2.27914554741916594442797353271, 2.88134115319943188416323233101, 2.93956834741724938854186651480, 3.58691966846064477060217541732, 3.72321739260956098462027395558, 4.22904794972117482123567903988, 4.23745593571510031171132037109, 4.27488739828705691199349531519, 4.53408477818591859292982505535, 5.11487609696476299226968965753, 5.28861207707756843996798095166, 5.53418972260422996179318487716, 5.58281611318310938060718697071, 5.79079032941488174358061833422, 6.12706998132374100489403724241, 6.24708437463745515654714362060, 6.40302877460966493467995695972, 6.79131042432532082199724315142, 6.89817361803044741509482742612, 6.97362593833939523540628652736
