| L(s) = 1 | − 4·19-s − 6·29-s − 20·31-s + 18·41-s + 13·49-s + 12·59-s − 2·61-s + 24·71-s + 20·79-s − 18·89-s − 36·101-s + 38·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 0.917·19-s − 1.11·29-s − 3.59·31-s + 2.81·41-s + 13/7·49-s + 1.56·59-s − 0.256·61-s + 2.84·71-s + 2.25·79-s − 1.90·89-s − 3.58·101-s + 3.63·109-s − 2·121-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 + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65610000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65610000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.058185619\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.058185619\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−7.85968408716017046476790503943, −7.65408368602153946910317893783, −7.34856309410404754744237377889, −6.97931461442362710704117522986, −6.71222020256871967185726484802, −6.26442757525536965973921745495, −5.79008505536397935651901477441, −5.61626185623178415255458117535, −5.23248344226306672608431661306, −5.05655537843348592630617836949, −4.22946755459864892833013698681, −3.99121276787776724876234885927, −3.87027439297491784311532031452, −3.47497600246573934504818003249, −2.72863547557977660385535780707, −2.44455112175389258535846772273, −1.98642216765270683645099395007, −1.69599074879575490912155350018, −0.878237822811585259886704335801, −0.39795018427201534517092136354,
0.39795018427201534517092136354, 0.878237822811585259886704335801, 1.69599074879575490912155350018, 1.98642216765270683645099395007, 2.44455112175389258535846772273, 2.72863547557977660385535780707, 3.47497600246573934504818003249, 3.87027439297491784311532031452, 3.99121276787776724876234885927, 4.22946755459864892833013698681, 5.05655537843348592630617836949, 5.23248344226306672608431661306, 5.61626185623178415255458117535, 5.79008505536397935651901477441, 6.26442757525536965973921745495, 6.71222020256871967185726484802, 6.97931461442362710704117522986, 7.34856309410404754744237377889, 7.65408368602153946910317893783, 7.85968408716017046476790503943
