| L(s) = 1 | − 2·3-s − 2·5-s + 2·7-s + 3·9-s − 8·13-s + 4·15-s − 2·17-s + 2·19-s − 4·21-s + 8·23-s − 4·25-s − 4·27-s − 2·29-s + 4·31-s − 4·35-s − 8·37-s + 16·39-s − 12·41-s − 6·45-s − 2·47-s + 3·49-s + 4·51-s + 14·53-s − 4·57-s + 8·59-s − 8·61-s + 6·63-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 0.755·7-s + 9-s − 2.21·13-s + 1.03·15-s − 0.485·17-s + 0.458·19-s − 0.872·21-s + 1.66·23-s − 4/5·25-s − 0.769·27-s − 0.371·29-s + 0.718·31-s − 0.676·35-s − 1.31·37-s + 2.56·39-s − 1.87·41-s − 0.894·45-s − 0.291·47-s + 3/7·49-s + 0.560·51-s + 1.92·53-s − 0.529·57-s + 1.04·59-s − 1.02·61-s + 0.755·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40755456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40755456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.076040245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.076040245\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 32 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 92 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 14 T + 152 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 10 T + 140 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T - 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 20 T + 246 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 172 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 202 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.072063092988033439727131666064, −7.75376337644143467283408474966, −7.40180112364703885069882109622, −7.07542486268669413064313404294, −6.93199094471312175867879323915, −6.64950284255965768803244196281, −6.00409022391890526108471468438, −5.57223473982031233498518546081, −5.23465130331779043845031751241, −5.03969043023911045835909285083, −4.58353450936060738986366608731, −4.55459432795056270482395860467, −3.79730279548064051110741373380, −3.60802639950858670481054499642, −2.93182901094852448434233079614, −2.57020932674238638439612322523, −1.83367136130467876464854791060, −1.71955712537866695108717160281, −0.68819536019624429886127449920, −0.42601561056753706415540454115,
0.42601561056753706415540454115, 0.68819536019624429886127449920, 1.71955712537866695108717160281, 1.83367136130467876464854791060, 2.57020932674238638439612322523, 2.93182901094852448434233079614, 3.60802639950858670481054499642, 3.79730279548064051110741373380, 4.55459432795056270482395860467, 4.58353450936060738986366608731, 5.03969043023911045835909285083, 5.23465130331779043845031751241, 5.57223473982031233498518546081, 6.00409022391890526108471468438, 6.64950284255965768803244196281, 6.93199094471312175867879323915, 7.07542486268669413064313404294, 7.40180112364703885069882109622, 7.75376337644143467283408474966, 8.072063092988033439727131666064
