| L(s) = 1 | − 2-s − 3-s + 3·5-s + 6-s + 7-s − 3·10-s − 11-s − 8·13-s − 14-s − 3·15-s + 12·17-s + 5·19-s − 21-s + 22-s + 10·23-s + 10·25-s + 8·26-s + 2·29-s + 3·30-s − 9·31-s + 32-s + 33-s − 12·34-s + 3·35-s + 12·37-s − 5·38-s + 8·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1.34·5-s + 0.408·6-s + 0.377·7-s − 0.948·10-s − 0.301·11-s − 2.21·13-s − 0.267·14-s − 0.774·15-s + 2.91·17-s + 1.14·19-s − 0.218·21-s + 0.213·22-s + 2.08·23-s + 2·25-s + 1.56·26-s + 0.371·29-s + 0.547·30-s − 1.61·31-s + 0.176·32-s + 0.174·33-s − 2.05·34-s + 0.507·35-s + 1.97·37-s − 0.811·38-s + 1.28·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.898345873\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.898345873\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 3 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 7 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 11 | $C_4$ | \( 1 + T - 9 T^{2} + p T^{3} + p^{2} T^{4} \) |
| good | 5 | $C_2^2:C_4$ | \( 1 - 3 T - T^{2} + 3 T^{3} + 16 T^{4} + 3 p T^{5} - p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 + 8 T + 51 T^{2} + 244 T^{3} + 1049 T^{4} + 244 p T^{5} + 51 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 12 T + 37 T^{2} + 180 T^{3} - 1619 T^{4} + 180 p T^{5} + 37 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 - 5 T - 9 T^{2} + 5 p T^{3} - 184 T^{4} + 5 p^{2} T^{5} - 9 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 5 T + 51 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2:C_4$ | \( 1 - 2 T + 35 T^{2} - 122 T^{3} + 689 T^{4} - 122 p T^{5} + 35 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 + 9 T + 75 T^{2} + 521 T^{3} + 3864 T^{4} + 521 p T^{5} + 75 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 - 12 T + 17 T^{2} + 360 T^{3} - 2879 T^{4} + 360 p T^{5} + 17 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 20 T + 149 T^{2} - 620 T^{3} + 2861 T^{4} - 620 p T^{5} + 149 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 14 T + 89 T^{2} + 798 T^{3} + 7529 T^{4} + 798 p T^{5} + 89 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 - 12 T + 91 T^{2} - 966 T^{3} + 9829 T^{4} - 966 p T^{5} + 91 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 18 T + 65 T^{2} - 1032 T^{3} - 14171 T^{4} - 1032 p T^{5} + 65 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_4$$\times$$C_4$ | \( ( 1 - T - 29 T^{2} - p T^{3} + p^{2} T^{4} )( 1 - T + 91 T^{2} - p T^{3} + p^{2} T^{4} ) \) |
| 67 | $D_{4}$ | \( ( 1 + 16 T + 178 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_4\times C_2$ | \( 1 - 12 T + 73 T^{2} - 24 T^{3} - 4895 T^{4} - 24 p T^{5} + 73 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 + 26 T + 203 T^{2} - 260 T^{3} - 12299 T^{4} - 260 p T^{5} + 203 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 - 8 T - 55 T^{2} + 482 T^{3} + 2149 T^{4} + 482 p T^{5} - 55 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 - 28 T + 461 T^{2} - 5964 T^{3} + 61769 T^{4} - 5964 p T^{5} + 461 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 17 T + 239 T^{2} - 17 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 24 T + 159 T^{2} - 248 T^{3} + 1089 T^{4} - 248 p T^{5} + 159 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.82867716897433269059540987152, −7.75097048847614403827328340877, −7.57726226296612252423081093367, −7.41914536495502405799971457176, −7.13579588075527485010325511451, −6.82858564873739929109556611259, −6.48073771798556650652968406000, −6.16623760046549232525674422471, −6.01403406816116959872561187917, −5.73106032887712450214033376579, −5.52167558654385592881398692068, −5.10578476128540057792553663594, −4.96066692486672050359356076451, −4.95005608990060540620901427307, −4.47591093323281900067620335310, −4.47296249641728526598337275123, −3.39017704483368188679900497137, −3.29212094750592263141566266927, −3.07747484074146045460086522563, −3.02592350621511411564294489224, −2.28112067473929980417813955229, −2.00837673792841754386614471899, −1.57811683828562166930062363741, −0.932030804009249173055129699981, −0.74649488008124585063646568320,
0.74649488008124585063646568320, 0.932030804009249173055129699981, 1.57811683828562166930062363741, 2.00837673792841754386614471899, 2.28112067473929980417813955229, 3.02592350621511411564294489224, 3.07747484074146045460086522563, 3.29212094750592263141566266927, 3.39017704483368188679900497137, 4.47296249641728526598337275123, 4.47591093323281900067620335310, 4.95005608990060540620901427307, 4.96066692486672050359356076451, 5.10578476128540057792553663594, 5.52167558654385592881398692068, 5.73106032887712450214033376579, 6.01403406816116959872561187917, 6.16623760046549232525674422471, 6.48073771798556650652968406000, 6.82858564873739929109556611259, 7.13579588075527485010325511451, 7.41914536495502405799971457176, 7.57726226296612252423081093367, 7.75097048847614403827328340877, 7.82867716897433269059540987152
