L(s) = 1 | + 2·5-s − 8·11-s − 8·19-s − 25-s + 4·29-s − 4·41-s + 10·49-s − 16·55-s + 24·59-s − 20·61-s + 16·71-s + 32·79-s + 12·89-s − 16·95-s − 12·101-s + 12·109-s + 26·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 2.41·11-s − 1.83·19-s − 1/5·25-s + 0.742·29-s − 0.624·41-s + 10/7·49-s − 2.15·55-s + 3.12·59-s − 2.56·61-s + 1.89·71-s + 3.60·79-s + 1.27·89-s − 1.64·95-s − 1.19·101-s + 1.14·109-s + 2.36·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.429543584\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.429543584\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−10.69624274799863726152397370424, −10.41957516066490583740359680763, −9.895211204366680163356715472970, −9.300063373497798212663900221466, −9.022982550101318757189098588511, −8.276342137901003229483989860351, −8.133884297769304140981235323564, −7.76827312540848623439299005793, −7.08880796671355039986654818950, −6.55522606158342277188410955736, −6.27952834674104816965489366041, −5.45469305988578750453821885930, −5.44294100118225694791529000095, −4.82460444005880207585720832866, −4.27430069994080419571574218214, −3.57907100035208770629193334516, −2.83190990143063948492180412278, −2.18795827357141964863149142728, −2.09572296176541280754924722822, −0.59036344855411467976633806951,
0.59036344855411467976633806951, 2.09572296176541280754924722822, 2.18795827357141964863149142728, 2.83190990143063948492180412278, 3.57907100035208770629193334516, 4.27430069994080419571574218214, 4.82460444005880207585720832866, 5.44294100118225694791529000095, 5.45469305988578750453821885930, 6.27952834674104816965489366041, 6.55522606158342277188410955736, 7.08880796671355039986654818950, 7.76827312540848623439299005793, 8.133884297769304140981235323564, 8.276342137901003229483989860351, 9.022982550101318757189098588511, 9.300063373497798212663900221466, 9.895211204366680163356715472970, 10.41957516066490583740359680763, 10.69624274799863726152397370424