| L(s) = 1 | + 16·7-s + 26·9-s − 28·17-s + 304·23-s + 138·25-s − 448·31-s − 140·41-s − 672·47-s − 494·49-s + 416·63-s + 144·71-s − 588·73-s + 928·79-s − 53·81-s + 532·89-s + 1.98e3·97-s − 2.35e3·103-s − 3.42e3·113-s − 448·119-s + 2.41e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 728·153-s + ⋯ |
| L(s) = 1 | + 0.863·7-s + 0.962·9-s − 0.399·17-s + 2.75·23-s + 1.10·25-s − 2.59·31-s − 0.533·41-s − 2.08·47-s − 1.44·49-s + 0.831·63-s + 0.240·71-s − 0.942·73-s + 1.32·79-s − 0.0727·81-s + 0.633·89-s + 2.08·97-s − 2.24·103-s − 2.84·113-s − 0.345·119-s + 1.81·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s − 0.384·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.588361089\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.588361089\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 26 T^{2} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 138 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 8 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 2410 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 1594 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 14 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 12346 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 152 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 23578 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 224 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 42058 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 70 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 33878 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 336 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 296746 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 125130 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 444890 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 571034 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 72 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 294 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 464 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 846522 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 266 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 994 T + p^{3} T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.36663995441058451432273437405, −16.31864788012553520013548286332, −15.22600646220450630565813851045, −14.75359118699136027758222501784, −14.56916251995215470821563962086, −13.38164339444501231594046082110, −12.97154344728769392956231004345, −12.56204586519946299862426039103, −11.42559369388467882930208929465, −11.05795936409614716996614663243, −10.48930337573917175152978976252, −9.432132866198387105181951896749, −8.962219872740643699774911562801, −8.064787010020897541921223108508, −7.17054636653215684856910850630, −6.70450696965871895365971855438, −5.20109583320908424907966408416, −4.72165569789740513650001362570, −3.32708994586277697613764332643, −1.55544138788419192924744643292,
1.55544138788419192924744643292, 3.32708994586277697613764332643, 4.72165569789740513650001362570, 5.20109583320908424907966408416, 6.70450696965871895365971855438, 7.17054636653215684856910850630, 8.064787010020897541921223108508, 8.962219872740643699774911562801, 9.432132866198387105181951896749, 10.48930337573917175152978976252, 11.05795936409614716996614663243, 11.42559369388467882930208929465, 12.56204586519946299862426039103, 12.97154344728769392956231004345, 13.38164339444501231594046082110, 14.56916251995215470821563962086, 14.75359118699136027758222501784, 15.22600646220450630565813851045, 16.31864788012553520013548286332, 16.36663995441058451432273437405
