L(s) = 1 | + 2-s + 3-s + 5·5-s + 6-s + 8·7-s + 5·10-s − 2·11-s − 6·13-s + 8·14-s + 5·15-s − 12·17-s + 8·21-s − 2·22-s − 6·23-s + 10·25-s − 6·26-s + 5·30-s − 12·31-s − 32-s − 2·33-s − 12·34-s + 40·35-s + 13·37-s − 6·39-s − 12·41-s + 8·42-s + 4·43-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 2.23·5-s + 0.408·6-s + 3.02·7-s + 1.58·10-s − 0.603·11-s − 1.66·13-s + 2.13·14-s + 1.29·15-s − 2.91·17-s + 1.74·21-s − 0.426·22-s − 1.25·23-s + 2·25-s − 1.17·26-s + 0.912·30-s − 2.15·31-s − 0.176·32-s − 0.348·33-s − 2.05·34-s + 6.76·35-s + 2.13·37-s − 0.960·39-s − 1.87·41-s + 1.23·42-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.507704728\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.507704728\) |
\(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 5 | $C_4$ | \( 1 - p T + 3 p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 11 | $C_4\times C_2$ | \( 1 + 2 T + 13 T^{2} + 34 T^{3} + 225 T^{4} + 34 p T^{5} + 13 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 + 6 T + 23 T^{2} + 120 T^{3} + 601 T^{4} + 120 p T^{5} + 23 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 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 | $C_2^2:C_4$ | \( 1 + 21 T^{2} - 10 T^{3} + 381 T^{4} - 10 p T^{5} + 21 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_4\times C_2$ | \( 1 + 6 T + 13 T^{2} - 60 T^{3} - 659 T^{4} - 60 p T^{5} + 13 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2:C_4$ | \( 1 - 19 T^{2} + 120 T^{3} + 721 T^{4} + 120 p T^{5} - 19 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 + 12 T + 113 T^{2} + 834 T^{3} + 5605 T^{4} + 834 p T^{5} + 113 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 - 13 T + 27 T^{2} + 445 T^{3} - 4264 T^{4} + 445 p T^{5} + 27 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 + 12 T + 53 T^{2} + 444 T^{3} + 4405 T^{4} + 444 p T^{5} + 53 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 2 T + 82 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 2 T + 17 T^{2} - 130 T^{3} + 761 T^{4} - 130 p T^{5} + 17 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 21 T + 253 T^{2} + 2535 T^{3} + 21196 T^{4} + 2535 p T^{5} + 253 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_4\times C_2$ | \( 1 - 20 T + 181 T^{2} - 1600 T^{3} + 14601 T^{4} - 1600 p T^{5} + 181 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 + 2 T + 3 T^{2} + 424 T^{3} + 4265 T^{4} + 424 p T^{5} + 3 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2:C_4$ | \( 1 - 18 T + 77 T^{2} - 30 T^{3} + 961 T^{4} - 30 p T^{5} + 77 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_4\times C_2$ | \( 1 - 28 T + 313 T^{2} - 2126 T^{3} + 14805 T^{4} - 2126 p T^{5} + 313 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 - 14 T + 3 T^{2} + 980 T^{3} - 9259 T^{4} + 980 p T^{5} + 3 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 83 | $C_4\times C_2$ | \( 1 + 6 T - 47 T^{2} - 780 T^{3} - 779 T^{4} - 780 p T^{5} - 47 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^2:C_4$ | \( 1 + 5 T - 79 T^{2} - 5 p T^{3} + 5276 T^{4} - 5 p^{2} T^{5} - 79 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 18 T + 47 T^{2} + 1740 T^{3} - 27059 T^{4} + 1740 p T^{5} + 47 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−9.687818995382257380037828020586, −9.243439402267130846148126510136, −8.967736304019538588441462549927, −8.881303827775368396817517740481, −8.499241886291921564136946711727, −8.095332281353903721876412967852, −7.76726723566014285915545147718, −7.76483694790135152754973126685, −7.64982664406823497799484814159, −6.86851847503758559561602606013, −6.52182546105273119245285082556, −6.36060986634045218351650888427, −6.31836009160644899111225545213, −5.31749180948017034830304535510, −5.28497309654397536401566236720, −5.18463910985769388165492696502, −4.98102964864389107454151966734, −4.56622908050813456726919366037, −4.21501948897060548742146284126, −3.88651645387973616330895132183, −3.16269405794208447701174193661, −2.39424812705518774258246690486, −2.13936383242117023427396859604, −1.89545622112054634888781633824, −1.89414489831281707489994724004,
1.89414489831281707489994724004, 1.89545622112054634888781633824, 2.13936383242117023427396859604, 2.39424812705518774258246690486, 3.16269405794208447701174193661, 3.88651645387973616330895132183, 4.21501948897060548742146284126, 4.56622908050813456726919366037, 4.98102964864389107454151966734, 5.18463910985769388165492696502, 5.28497309654397536401566236720, 5.31749180948017034830304535510, 6.31836009160644899111225545213, 6.36060986634045218351650888427, 6.52182546105273119245285082556, 6.86851847503758559561602606013, 7.64982664406823497799484814159, 7.76483694790135152754973126685, 7.76726723566014285915545147718, 8.095332281353903721876412967852, 8.499241886291921564136946711727, 8.881303827775368396817517740481, 8.967736304019538588441462549927, 9.243439402267130846148126510136, 9.687818995382257380037828020586