| L(s) = 1 | + 4·5-s − 10·13-s − 10·17-s + 11·25-s + 10·37-s + 16·41-s + 10·53-s − 24·61-s − 40·65-s + 10·73-s − 9·81-s − 40·85-s + 10·97-s + 4·101-s + 30·113-s + 22·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 50·169-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 2.77·13-s − 2.42·17-s + 11/5·25-s + 1.64·37-s + 2.49·41-s + 1.37·53-s − 3.07·61-s − 4.96·65-s + 1.17·73-s − 81-s − 4.33·85-s + 1.01·97-s + 0.398·101-s + 2.82·113-s + 2·121-s + 2.14·125-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 + 3.84·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.028080073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.028080073\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−14.36196059830176043760838360071, −14.32719116835131344129120718687, −13.52087354184667345390636775401, −13.13304650984422156221169451965, −12.60039612390521967293181515483, −12.18903620283370482013499177600, −11.22182095132037467626831895479, −10.85059261549145160635221462498, −10.06377174762825158964122584652, −9.674845215232649046593868163176, −9.215903905955930506680129574560, −8.755222109170443434349305797563, −7.53894588705916063426304420155, −7.18411973635644872290120913485, −6.29307059904154622423476335696, −5.87142516828485734671431238577, −4.75363631493516026702961808663, −4.59227006210364649411590411646, −2.49781743409028979957278186652, −2.33847888997287159193861359754,
2.33847888997287159193861359754, 2.49781743409028979957278186652, 4.59227006210364649411590411646, 4.75363631493516026702961808663, 5.87142516828485734671431238577, 6.29307059904154622423476335696, 7.18411973635644872290120913485, 7.53894588705916063426304420155, 8.755222109170443434349305797563, 9.215903905955930506680129574560, 9.674845215232649046593868163176, 10.06377174762825158964122584652, 10.85059261549145160635221462498, 11.22182095132037467626831895479, 12.18903620283370482013499177600, 12.60039612390521967293181515483, 13.13304650984422156221169451965, 13.52087354184667345390636775401, 14.32719116835131344129120718687, 14.36196059830176043760838360071
