| L(s) = 1 | − 2·2-s + 6·3-s + 2·4-s − 2·5-s − 12·6-s − 2·8-s + 21·9-s + 4·10-s + 4·11-s + 12·12-s − 10·13-s − 12·15-s + 3·16-s − 16·17-s − 42·18-s + 2·19-s − 4·20-s − 8·22-s + 10·23-s − 12·24-s + 5·25-s + 20·26-s + 54·27-s − 24·29-s + 24·30-s − 8·32-s + 24·33-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3.46·3-s + 4-s − 0.894·5-s − 4.89·6-s − 0.707·8-s + 7·9-s + 1.26·10-s + 1.20·11-s + 3.46·12-s − 2.77·13-s − 3.09·15-s + 3/4·16-s − 3.88·17-s − 9.89·18-s + 0.458·19-s − 0.894·20-s − 1.70·22-s + 2.08·23-s − 2.44·24-s + 25-s + 3.92·26-s + 10.3·27-s − 4.45·29-s + 4.38·30-s − 1.41·32-s + 4.17·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.845569675\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.845569675\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 + p T + p T^{2} + p T^{3} + T^{4} + p^{2} T^{5} + p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 10 T + 2 p T^{2} - 108 T^{3} - 841 T^{4} - 108 p T^{5} + 2 p^{3} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 16 T + 80 T^{2} - 8 T^{3} - 1121 T^{4} - 8 p T^{5} + 80 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 2 T - 32 T^{2} + 4 T^{3} + 859 T^{4} + 4 p T^{5} - 32 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 10 T + 74 T^{2} - 400 T^{3} + 2071 T^{4} - 400 p T^{5} + 74 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 24 T + 289 T^{2} + 2328 T^{3} + 14136 T^{4} + 2328 p T^{5} + 289 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 3846 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 6 T + 90 T^{2} + 552 T^{3} + 4439 T^{4} + 552 p T^{5} + 90 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^3$ | \( 1 + 73 T^{2} + 3648 T^{4} + 73 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 6 T + 9 T^{2} + 6 p T^{3} - 2872 T^{4} + 6 p^{2} T^{5} + 9 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 18 T + 162 T^{2} - 1548 T^{3} + 13271 T^{4} - 1548 p T^{5} + 162 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 8 T + 20 T^{2} + 196 T^{3} - 3641 T^{4} + 196 p T^{5} + 20 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 6 T - 20 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 7734 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 180 T^{3} + 3566 T^{4} + 180 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 24 T + 144 T^{2} - 24 p T^{3} - 31057 T^{4} - 24 p^{2} T^{5} + 144 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 16134 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 26 T + 365 T^{2} - 3554 T^{3} + 32320 T^{4} - 3554 p T^{5} + 365 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 16 T + 62 T^{2} - 256 T^{3} + 6931 T^{4} - 256 p T^{5} + 62 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 34 T + 458 T^{2} + 2916 T^{3} + 15023 T^{4} + 2916 p T^{5} + 458 p^{2} T^{6} + 34 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.645424227613220889329598356230, −8.392009766172471380148874762744, −8.061075703228973972945944961678, −7.63106138273762640400683243919, −7.59401869122062962830019246519, −7.43143544960941044524187427340, −6.99381119881990877093806561021, −6.97025666109637453231350398094, −6.83493990942902842565012806328, −6.81923272892507492237727041591, −5.79675996674074115427596972682, −5.64562113995676952097901305651, −5.19930458325914388691063733844, −4.62946203709995138081258132328, −4.53910515989732264836057147150, −4.19816802089695976760273790875, −4.05067203139981384943874012552, −3.43521235228360260303980207587, −3.41701041013125463758644653766, −3.00939325994860998949424756780, −2.47502258876507064144677674215, −2.22589185022683709723396167373, −1.89712901819148144537649828625, −1.88341844662801733029715740647, −0.57604045978410033875167891103,
0.57604045978410033875167891103, 1.88341844662801733029715740647, 1.89712901819148144537649828625, 2.22589185022683709723396167373, 2.47502258876507064144677674215, 3.00939325994860998949424756780, 3.41701041013125463758644653766, 3.43521235228360260303980207587, 4.05067203139981384943874012552, 4.19816802089695976760273790875, 4.53910515989732264836057147150, 4.62946203709995138081258132328, 5.19930458325914388691063733844, 5.64562113995676952097901305651, 5.79675996674074115427596972682, 6.81923272892507492237727041591, 6.83493990942902842565012806328, 6.97025666109637453231350398094, 6.99381119881990877093806561021, 7.43143544960941044524187427340, 7.59401869122062962830019246519, 7.63106138273762640400683243919, 8.061075703228973972945944961678, 8.392009766172471380148874762744, 8.645424227613220889329598356230
