| L(s) = 1 | − 7-s + 3·9-s − 10·11-s + 12·23-s + 25-s − 18·29-s + 4·37-s + 20·43-s + 49-s + 8·53-s − 3·63-s + 24·67-s + 16·71-s + 10·77-s + 26·79-s − 30·99-s − 12·107-s − 6·109-s + 28·113-s + 53·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 9-s − 3.01·11-s + 2.50·23-s + 1/5·25-s − 3.34·29-s + 0.657·37-s + 3.04·43-s + 1/7·49-s + 1.09·53-s − 0.377·63-s + 2.93·67-s + 1.89·71-s + 1.13·77-s + 2.92·79-s − 3.01·99-s − 1.16·107-s − 0.574·109-s + 2.63·113-s + 4.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 137200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.265506963\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.265506963\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
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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
−9.399157544602484082257478460833, −9.025895505037624522078550270792, −8.254601233250958389865237776453, −7.73731776455161087713955528993, −7.33986466138726521878856236101, −7.23579943557103217993182216757, −6.38103485258561037052446742413, −5.62823751414852416971700636779, −5.11248277682454093810005074849, −5.09318860647436101841510990708, −4.00033630632803641873619633981, −3.51475005035302395747602864778, −2.51163217316890307126702952381, −2.33062913295078547814440271884, −0.76376003838796265827370008029,
0.76376003838796265827370008029, 2.33062913295078547814440271884, 2.51163217316890307126702952381, 3.51475005035302395747602864778, 4.00033630632803641873619633981, 5.09318860647436101841510990708, 5.11248277682454093810005074849, 5.62823751414852416971700636779, 6.38103485258561037052446742413, 7.23579943557103217993182216757, 7.33986466138726521878856236101, 7.73731776455161087713955528993, 8.254601233250958389865237776453, 9.025895505037624522078550270792, 9.399157544602484082257478460833
