| L(s) = 1 | + 2-s + 3·3-s − 2·4-s − 3·5-s + 3·6-s + 7-s − 3·8-s + 2·9-s − 3·10-s − 6·11-s − 6·12-s − 2·13-s + 14-s − 9·15-s + 16-s + 11·17-s + 2·18-s − 10·19-s + 6·20-s + 3·21-s − 6·22-s + 3·23-s − 9·24-s − 2·25-s − 2·26-s − 6·27-s − 2·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s − 4-s − 1.34·5-s + 1.22·6-s + 0.377·7-s − 1.06·8-s + 2/3·9-s − 0.948·10-s − 1.80·11-s − 1.73·12-s − 0.554·13-s + 0.267·14-s − 2.32·15-s + 1/4·16-s + 2.66·17-s + 0.471·18-s − 2.29·19-s + 1.34·20-s + 0.654·21-s − 1.27·22-s + 0.625·23-s − 1.83·24-s − 2/5·25-s − 0.392·26-s − 1.15·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38950081 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38950081 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.205702566\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.205702566\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 79 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - p T + 7 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 3 T + 11 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 11 T + 63 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 37 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5 T + 63 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T + 51 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 11 T + 93 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 57 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 113 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 19 T + 231 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 26 T + 358 T^{2} - 26 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.088808151867624468544295779242, −7.958131402797158765766542181149, −7.64493377311528016087572747355, −7.54881107888317084459926290306, −7.16636808075744020255092437890, −6.21331257083505664636071903372, −5.90337606939955348661065376132, −5.82866504633401957571597358339, −5.05670535390183131945891293442, −4.95337157483340315980570945109, −4.39657804293854026344213834161, −4.28800070971414002470202254061, −3.79064233470699910673004490078, −3.39001132239993464937032018775, −3.08401827971220217655989417766, −2.83114477161929139361069925869, −2.14371542346475012584752962748, −2.09576741850591639257631148822, −0.73512015694293003977430604477, −0.51607231554867512018478253512,
0.51607231554867512018478253512, 0.73512015694293003977430604477, 2.09576741850591639257631148822, 2.14371542346475012584752962748, 2.83114477161929139361069925869, 3.08401827971220217655989417766, 3.39001132239993464937032018775, 3.79064233470699910673004490078, 4.28800070971414002470202254061, 4.39657804293854026344213834161, 4.95337157483340315980570945109, 5.05670535390183131945891293442, 5.82866504633401957571597358339, 5.90337606939955348661065376132, 6.21331257083505664636071903372, 7.16636808075744020255092437890, 7.54881107888317084459926290306, 7.64493377311528016087572747355, 7.958131402797158765766542181149, 8.088808151867624468544295779242
