| L(s) = 1 | + 3-s + 9-s + 8·11-s + 6·25-s + 27-s + 8·33-s + 14·49-s − 8·59-s − 20·73-s + 6·75-s + 81-s − 8·83-s + 4·97-s + 8·99-s + 24·107-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 14·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 2.41·11-s + 6/5·25-s + 0.192·27-s + 1.39·33-s + 2·49-s − 1.04·59-s − 2.34·73-s + 0.692·75-s + 1/9·81-s − 0.878·83-s + 0.406·97-s + 0.804·99-s + 2.32·107-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.15·147-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.968248410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.968248410\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.687836611869498461248501001358, −8.383770290277106663524379844615, −7.44186433189479833640188644464, −7.32441608908158846095152437813, −6.81858341510728817091787808945, −6.26442416232961782239990972473, −5.99905451357318802687535645927, −5.26894217635180204856173110634, −4.48078333578381963710620773492, −4.29756935854961224319348061414, −3.63509414118325605441393326947, −3.18318808988312733225752583400, −2.44766734657440010134892551021, −1.59682305302508404492342801678, −1.02275759495493634545990611503,
1.02275759495493634545990611503, 1.59682305302508404492342801678, 2.44766734657440010134892551021, 3.18318808988312733225752583400, 3.63509414118325605441393326947, 4.29756935854961224319348061414, 4.48078333578381963710620773492, 5.26894217635180204856173110634, 5.99905451357318802687535645927, 6.26442416232961782239990972473, 6.81858341510728817091787808945, 7.32441608908158846095152437813, 7.44186433189479833640188644464, 8.383770290277106663524379844615, 8.687836611869498461248501001358
