| L(s) = 1 | − 4·4-s + 8·5-s + 24·13-s + 16·16-s + 46·17-s − 32·20-s + 39·25-s − 140·37-s + 98·41-s + 98·49-s − 96·52-s − 146·53-s + 240·61-s − 64·64-s + 192·65-s − 184·68-s + 128·80-s − 81·81-s + 368·85-s + 82·89-s − 156·100-s − 302·109-s − 194·113-s + 112·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 4-s + 8/5·5-s + 1.84·13-s + 16-s + 2.70·17-s − 8/5·20-s + 1.55·25-s − 3.78·37-s + 2.39·41-s + 2·49-s − 1.84·52-s − 2.75·53-s + 3.93·61-s − 64-s + 2.95·65-s − 2.70·68-s + 8/5·80-s − 81-s + 4.32·85-s + 0.921·89-s − 1.55·100-s − 2.77·109-s − 1.71·113-s + 0.895·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.818672925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.818672925\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | $C_2$ | \( 1 - 8 T + p^{2} T^{2} \) |
| 13 | $C_2$ | \( 1 - 24 T + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 11 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 30 T + p^{2} T^{2} )( 1 - 16 T + p^{2} T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 40 T + p^{2} T^{2} )( 1 + 40 T + p^{2} T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 70 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 80 T + p^{2} T^{2} )( 1 - 18 T + p^{2} T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 56 T + p^{2} T^{2} )( 1 + 90 T + p^{2} T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 120 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 110 T + p^{2} T^{2} )( 1 + 110 T + p^{2} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 160 T + p^{2} T^{2} )( 1 + 78 T + p^{2} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 144 T + p^{2} T^{2} )( 1 + 144 T + p^{2} T^{2} ) \) |
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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
−12.43768299013012723554228705701, −11.54169370295361309883336887426, −10.73948544399253604405578215282, −10.54793473939567549907093036261, −9.981352072860518166058337415805, −9.727438736998799721617435516179, −9.080500563412974253712248163094, −8.789945033372999982205647882775, −8.239820399867512364993634439988, −7.73485263856560258139759625831, −6.99669276201155875138412440017, −6.29050153310738294463201414182, −5.80193834206659892438619275757, −5.32669321763289190593946926986, −5.18195813625108048702010514335, −3.79372153408123126865611341342, −3.69930554269742571616760706159, −2.74305640833824121925391702786, −1.51702449899746355977099453353, −1.04590292577908300176184117453,
1.04590292577908300176184117453, 1.51702449899746355977099453353, 2.74305640833824121925391702786, 3.69930554269742571616760706159, 3.79372153408123126865611341342, 5.18195813625108048702010514335, 5.32669321763289190593946926986, 5.80193834206659892438619275757, 6.29050153310738294463201414182, 6.99669276201155875138412440017, 7.73485263856560258139759625831, 8.239820399867512364993634439988, 8.789945033372999982205647882775, 9.080500563412974253712248163094, 9.727438736998799721617435516179, 9.981352072860518166058337415805, 10.54793473939567549907093036261, 10.73948544399253604405578215282, 11.54169370295361309883336887426, 12.43768299013012723554228705701
