| L(s) = 1 | − 3·9-s − 16·19-s − 10·25-s + 20·37-s + 2·49-s + 16·67-s + 20·73-s + 9·81-s − 40·103-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + 48·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 9-s − 3.67·19-s − 2·25-s + 3.28·37-s + 2/7·49-s + 1.95·67-s + 2.34·73-s + 81-s − 3.94·103-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 − 1.69·169-s + 3.67·171-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 646416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 646416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9360834884\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9360834884\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 67 | $C_2$ | \( 1 - 16 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−10.58239841310948717861466150815, −10.03707561523877332423017607821, −9.586297368693113614199599699184, −9.175268508548791978029539178588, −8.837342018888919708482711389547, −8.119684511693085169170049208515, −7.957295659170044267405941876662, −7.900483632880569114871967100450, −6.75108053456014367237310928742, −6.41887827763066451713508270105, −6.32190872347581637056997581125, −5.60215186231362226963792820265, −5.29734093237376108657972354062, −4.40960358390524532506210086186, −4.05669264461900204148751515068, −3.83822977238911175542549080149, −2.62395235524426326158758143451, −2.47947985292924418226681639704, −1.81510236695420132804793374849, −0.45889908113112294747661316146,
0.45889908113112294747661316146, 1.81510236695420132804793374849, 2.47947985292924418226681639704, 2.62395235524426326158758143451, 3.83822977238911175542549080149, 4.05669264461900204148751515068, 4.40960358390524532506210086186, 5.29734093237376108657972354062, 5.60215186231362226963792820265, 6.32190872347581637056997581125, 6.41887827763066451713508270105, 6.75108053456014367237310928742, 7.900483632880569114871967100450, 7.957295659170044267405941876662, 8.119684511693085169170049208515, 8.837342018888919708482711389547, 9.175268508548791978029539178588, 9.586297368693113614199599699184, 10.03707561523877332423017607821, 10.58239841310948717861466150815
