| L(s) = 1 | + 2.73·2-s − 0.524·4-s + 21.1·5-s − 25.8·7-s − 23.3·8-s + 57.7·10-s + 6.96·11-s − 70.7·14-s − 59.5·16-s − 122.·17-s + 43.1·19-s − 11.0·20-s + 19.0·22-s + 75.5·23-s + 321.·25-s + 13.5·28-s + 163.·29-s + 139.·31-s + 23.6·32-s − 335.·34-s − 546.·35-s + 2.80·37-s + 117.·38-s − 492.·40-s + 300.·41-s + 363.·43-s − 3.65·44-s + ⋯ |
| L(s) = 1 | + 0.966·2-s − 0.0655·4-s + 1.88·5-s − 1.39·7-s − 1.03·8-s + 1.82·10-s + 0.191·11-s − 1.34·14-s − 0.930·16-s − 1.75·17-s + 0.520·19-s − 0.123·20-s + 0.184·22-s + 0.684·23-s + 2.56·25-s + 0.0915·28-s + 1.04·29-s + 0.807·31-s + 0.130·32-s − 1.69·34-s − 2.63·35-s + 0.0124·37-s + 0.503·38-s − 1.94·40-s + 1.14·41-s + 1.28·43-s − 0.0125·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.663619637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.663619637\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.73T + 8T^{2} \) |
| 5 | \( 1 - 21.1T + 125T^{2} \) |
| 7 | \( 1 + 25.8T + 343T^{2} \) |
| 11 | \( 1 - 6.96T + 1.33e3T^{2} \) |
| 17 | \( 1 + 122.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 43.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 75.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 163.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 139.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 2.80T + 5.06e4T^{2} \) |
| 41 | \( 1 - 300.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 363.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 41.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 125.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 407.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 536.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 340.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 514.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 491.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 762.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 345.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 362.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 276.T + 9.12e5T^{2} \) |
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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
−9.177582355177815638685854769657, −8.749116198248757728968006728636, −6.90613432559055292133097151658, −6.41467534798716632732270283247, −5.85751324945471705883807886179, −5.02913548859681649952491497045, −4.12659921717184520656399400494, −2.88598817862226020722546152245, −2.41741290519743312791190638801, −0.796244676971123835000639304920,
0.796244676971123835000639304920, 2.41741290519743312791190638801, 2.88598817862226020722546152245, 4.12659921717184520656399400494, 5.02913548859681649952491497045, 5.85751324945471705883807886179, 6.41467534798716632732270283247, 6.90613432559055292133097151658, 8.749116198248757728968006728636, 9.177582355177815638685854769657
