| L(s) = 1 | + 2·5-s − 2·11-s + 6·19-s − 25-s − 6·29-s + 2·31-s − 2·41-s + 10·49-s − 4·55-s − 26·59-s − 20·61-s − 18·71-s + 26·89-s + 12·95-s + 2·101-s + 34·109-s − 19·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s − 12·145-s + 149-s + 151-s + 4·155-s + 157-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.603·11-s + 1.37·19-s − 1/5·25-s − 1.11·29-s + 0.359·31-s − 0.312·41-s + 10/7·49-s − 0.539·55-s − 3.38·59-s − 2.56·61-s − 2.13·71-s + 2.75·89-s + 1.23·95-s + 0.199·101-s + 3.25·109-s − 1.72·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.996·145-s + 0.0819·149-s + 0.0813·151-s + 0.321·155-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.206572543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.206572543\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−9.402518046033215876007521806451, −9.210834644242579827099640457259, −9.081867706661749262303703922302, −8.440391139788530407502011339614, −7.82376424890420270715264499109, −7.55356728093668103586015384346, −7.42845618731090441428154377970, −6.78854792910570009001612938496, −6.16163975076308164619575841609, −5.85110325422455676325086727041, −5.75704678840925281128224618964, −4.97671650461894194341323598667, −4.77810163572002093338866786937, −4.22952627523319180085593402047, −3.49654634105061005795087374782, −3.07217952417058175831613800830, −2.70949840514213828956648547999, −1.77428118372607824160136503159, −1.66522220777759512472939893383, −0.56370449103362275884195960392,
0.56370449103362275884195960392, 1.66522220777759512472939893383, 1.77428118372607824160136503159, 2.70949840514213828956648547999, 3.07217952417058175831613800830, 3.49654634105061005795087374782, 4.22952627523319180085593402047, 4.77810163572002093338866786937, 4.97671650461894194341323598667, 5.75704678840925281128224618964, 5.85110325422455676325086727041, 6.16163975076308164619575841609, 6.78854792910570009001612938496, 7.42845618731090441428154377970, 7.55356728093668103586015384346, 7.82376424890420270715264499109, 8.440391139788530407502011339614, 9.081867706661749262303703922302, 9.210834644242579827099640457259, 9.402518046033215876007521806451
