| L(s) = 1 | − 4·2-s + 10·4-s − 20·8-s + 35·16-s + 16·17-s − 4·19-s + 8·23-s − 16·31-s − 56·32-s − 64·34-s + 16·38-s − 32·46-s + 24·47-s − 12·49-s + 24·53-s − 8·61-s + 64·62-s + 84·64-s + 160·68-s − 40·76-s − 16·79-s + 24·83-s + 80·92-s − 96·94-s + 48·98-s − 96·106-s + 16·107-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 5·4-s − 7.07·8-s + 35/4·16-s + 3.88·17-s − 0.917·19-s + 1.66·23-s − 2.87·31-s − 9.89·32-s − 10.9·34-s + 2.59·38-s − 4.71·46-s + 3.50·47-s − 1.71·49-s + 3.29·53-s − 1.02·61-s + 8.12·62-s + 21/2·64-s + 19.4·68-s − 4.58·76-s − 1.80·79-s + 2.63·83-s + 8.34·92-s − 9.90·94-s + 4.84·98-s − 9.32·106-s + 1.54·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.176019202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.176019202\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 12 T^{2} + 86 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 28 T^{2} + 390 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 36 T^{2} + 614 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4$ | \( 1 + 4 T^{2} + 486 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 100 T^{2} + 4806 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 116 T^{2} + 6294 T^{4} + 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 124 T^{2} + 7110 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 59 | $D_4\times C_2$ | \( 1 + 28 T^{2} + 4086 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 67 | $D_4\times C_2$ | \( 1 + 76 T^{2} + 3510 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 28 T^{2} - 2010 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 100 T^{2} + 6246 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 83 | $D_{4}$ | \( ( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 340 T^{2} + 44694 T^{4} + 340 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 84 T^{2} - 586 T^{4} + 84 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−5.55722437969964745652526151361, −5.21031952003638155351156289916, −5.17941577992912746892920300396, −5.14254902782264809448903150155, −5.10441394555074785725631513717, −4.36259758988656949720918449796, −4.23328901518856088524296525652, −4.05361996922746835819714381039, −3.96899280829558807197665597778, −3.53530677900477817828254384770, −3.39338429445484526306101826343, −3.38153697078452403774993487720, −3.13028020952302344928969142925, −2.77225833379982273298216925808, −2.67528240211387757488706279062, −2.45515367166318905173218956352, −2.34280464927140107913830164149, −1.72101642526050843118719793769, −1.61634577728634965090589318241, −1.57849039579225687214878824750, −1.57363795888864509800940876943, −0.77299161510418424374884988288, −0.72609050203460999447952007492, −0.60993193400425563090239861242, −0.50832012925291728711442672993,
0.50832012925291728711442672993, 0.60993193400425563090239861242, 0.72609050203460999447952007492, 0.77299161510418424374884988288, 1.57363795888864509800940876943, 1.57849039579225687214878824750, 1.61634577728634965090589318241, 1.72101642526050843118719793769, 2.34280464927140107913830164149, 2.45515367166318905173218956352, 2.67528240211387757488706279062, 2.77225833379982273298216925808, 3.13028020952302344928969142925, 3.38153697078452403774993487720, 3.39338429445484526306101826343, 3.53530677900477817828254384770, 3.96899280829558807197665597778, 4.05361996922746835819714381039, 4.23328901518856088524296525652, 4.36259758988656949720918449796, 5.10441394555074785725631513717, 5.14254902782264809448903150155, 5.17941577992912746892920300396, 5.21031952003638155351156289916, 5.55722437969964745652526151361
