| L(s) = 1 | + 2·5-s − 2·11-s − 2·13-s + 2·17-s + 2·19-s + 2·23-s + 25-s + 2·41-s − 2·43-s − 4·55-s − 4·65-s + 4·85-s + 4·95-s − 2·103-s + 2·107-s − 2·113-s + 4·115-s + 121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 4·143-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 2·5-s − 2·11-s − 2·13-s + 2·17-s + 2·19-s + 2·23-s + 25-s + 2·41-s − 2·43-s − 4·55-s − 4·65-s + 4·85-s + 4·95-s − 2·103-s + 2·107-s − 2·113-s + 4·115-s + 121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 4·143-s + 149-s + 151-s + 157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5992704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5992704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.789913009\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.789913009\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + T^{4} \) |
| 37 | $C_2^2$ | \( 1 + T^{4} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + T^{4} \) |
| 53 | $C_2^2$ | \( 1 + T^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2^2$ | \( 1 + T^{4} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2^2$ | \( 1 + T^{4} \) |
| 89 | $C_2^2$ | \( 1 + T^{4} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−9.411235665789165665742062316307, −9.318673562884599759903396227912, −8.575258828568326335319606212382, −7.916638521008225445748590472741, −7.75843990644232634191827378766, −7.56772049893492998970091342130, −6.92309105069836742310596638135, −6.84418102389255540213900169227, −5.92928189347892353234292917237, −5.72283844985545569111326160417, −5.26446251428881365846986867713, −5.21274652103574560327797028027, −4.97825624274766703740845303727, −4.25745655066536670597549280370, −3.18506183361364499007532921764, −3.13103680615111031200272645169, −2.64983340469225803456731695853, −2.25157484280692297512002382403, −1.59156029747143447406200192966, −0.939053410406185639029433388456,
0.939053410406185639029433388456, 1.59156029747143447406200192966, 2.25157484280692297512002382403, 2.64983340469225803456731695853, 3.13103680615111031200272645169, 3.18506183361364499007532921764, 4.25745655066536670597549280370, 4.97825624274766703740845303727, 5.21274652103574560327797028027, 5.26446251428881365846986867713, 5.72283844985545569111326160417, 5.92928189347892353234292917237, 6.84418102389255540213900169227, 6.92309105069836742310596638135, 7.56772049893492998970091342130, 7.75843990644232634191827378766, 7.916638521008225445748590472741, 8.575258828568326335319606212382, 9.318673562884599759903396227912, 9.411235665789165665742062316307
