| L(s) = 1 | + 2·3-s − 3·9-s + 4·13-s + 6·17-s + 12·23-s + 25-s − 14·27-s + 8·39-s + 2·43-s + 5·49-s + 12·51-s − 12·53-s − 16·61-s + 24·69-s + 2·75-s − 20·79-s − 4·81-s + 24·101-s − 28·103-s + 24·107-s − 12·113-s − 12·117-s + 22·121-s + 127-s + 4·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 9-s + 1.10·13-s + 1.45·17-s + 2.50·23-s + 1/5·25-s − 2.69·27-s + 1.28·39-s + 0.304·43-s + 5/7·49-s + 1.68·51-s − 1.64·53-s − 2.04·61-s + 2.88·69-s + 0.230·75-s − 2.25·79-s − 4/9·81-s + 2.38·101-s − 2.75·103-s + 2.32·107-s − 1.12·113-s − 1.10·117-s + 2·121-s + 0.0887·127-s + 0.352·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.892784823\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.892784823\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 83 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−12.56591945348220648660290116662, −12.34696847050935546684804652172, −11.34913759877931000882715344370, −11.34321801490786955910351958186, −10.79841486861687876145581655720, −10.15131982791510938619243267869, −9.455261620798328003273195771539, −9.085027294343770777167117739192, −8.621273572444831610393501303142, −8.382982560575028275442200366894, −7.55056661700761708834136011260, −7.39812493034566573189772641080, −6.36392121518998361378249381242, −5.88295087585444615924522111830, −5.33610604815166923439313883019, −4.59575373214872018470438913342, −3.48628648044729452992003112336, −3.22516482017092836675604231153, −2.64144545797811667872857088370, −1.32750074296120482138943209608,
1.32750074296120482138943209608, 2.64144545797811667872857088370, 3.22516482017092836675604231153, 3.48628648044729452992003112336, 4.59575373214872018470438913342, 5.33610604815166923439313883019, 5.88295087585444615924522111830, 6.36392121518998361378249381242, 7.39812493034566573189772641080, 7.55056661700761708834136011260, 8.382982560575028275442200366894, 8.621273572444831610393501303142, 9.085027294343770777167117739192, 9.455261620798328003273195771539, 10.15131982791510938619243267869, 10.79841486861687876145581655720, 11.34321801490786955910351958186, 11.34913759877931000882715344370, 12.34696847050935546684804652172, 12.56591945348220648660290116662
