| L(s) = 1 | + 2.56·3-s + 5-s + 7-s + 3.56·9-s + 1.43·11-s − 0.561·13-s + 2.56·15-s + 5.68·17-s + 4·19-s + 2.56·21-s + 25-s + 1.43·27-s − 4.56·29-s − 5.12·31-s + 3.68·33-s + 35-s − 0.876·37-s − 1.43·39-s − 8.24·41-s + 3.56·45-s − 6.56·47-s + 49-s + 14.5·51-s + 7.12·53-s + 1.43·55-s + 10.2·57-s + 14.2·59-s + ⋯ |
| L(s) = 1 | + 1.47·3-s + 0.447·5-s + 0.377·7-s + 1.18·9-s + 0.433·11-s − 0.155·13-s + 0.661·15-s + 1.37·17-s + 0.917·19-s + 0.558·21-s + 0.200·25-s + 0.276·27-s − 0.847·29-s − 0.920·31-s + 0.641·33-s + 0.169·35-s − 0.144·37-s − 0.230·39-s − 1.28·41-s + 0.530·45-s − 0.957·47-s + 0.142·49-s + 2.03·51-s + 0.978·53-s + 0.193·55-s + 1.35·57-s + 1.85·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.663129189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.663129189\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 - 2.56T + 3T^{2} \) |
| 11 | \( 1 - 1.43T + 11T^{2} \) |
| 13 | \( 1 + 0.561T + 13T^{2} \) |
| 17 | \( 1 - 5.68T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 4.56T + 29T^{2} \) |
| 31 | \( 1 + 5.12T + 31T^{2} \) |
| 37 | \( 1 + 0.876T + 37T^{2} \) |
| 41 | \( 1 + 8.24T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 6.56T + 47T^{2} \) |
| 53 | \( 1 - 7.12T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 - 8.24T + 61T^{2} \) |
| 67 | \( 1 + 2.24T + 67T^{2} \) |
| 71 | \( 1 - 2.24T + 71T^{2} \) |
| 73 | \( 1 + 8.24T + 73T^{2} \) |
| 79 | \( 1 - 1.43T + 79T^{2} \) |
| 83 | \( 1 + 6.24T + 83T^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 97 | \( 1 - 5.68T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.898138455089126478920686583624, −8.414609223930360133552256839873, −7.53758490949812495422541699678, −7.07182338298911535217545864315, −5.77730121391923839375213494687, −5.08093024326893826063752364715, −3.80311069520315656278113327687, −3.27656861941984304829516364632, −2.21892671894437163033372355804, −1.34134377915277455325161836655,
1.34134377915277455325161836655, 2.21892671894437163033372355804, 3.27656861941984304829516364632, 3.80311069520315656278113327687, 5.08093024326893826063752364715, 5.77730121391923839375213494687, 7.07182338298911535217545864315, 7.53758490949812495422541699678, 8.414609223930360133552256839873, 8.898138455089126478920686583624
