| L(s) = 1 | − 3.23·2-s − 3·3-s + 2.47·4-s − 13.5·5-s + 9.70·6-s + 1.42·7-s + 17.8·8-s + 9·9-s + 43.9·10-s + 54.5·11-s − 7.42·12-s − 4.62·14-s + 40.7·15-s − 77.6·16-s − 114.·17-s − 29.1·18-s − 104.·19-s − 33.6·20-s − 4.28·21-s − 176.·22-s − 64.5·23-s − 53.6·24-s + 59.4·25-s − 27·27-s + 3.53·28-s − 60.8·29-s − 131.·30-s + ⋯ |
| L(s) = 1 | − 1.14·2-s − 0.577·3-s + 0.309·4-s − 1.21·5-s + 0.660·6-s + 0.0771·7-s + 0.790·8-s + 0.333·9-s + 1.39·10-s + 1.49·11-s − 0.178·12-s − 0.0883·14-s + 0.701·15-s − 1.21·16-s − 1.63·17-s − 0.381·18-s − 1.26·19-s − 0.375·20-s − 0.0445·21-s − 1.71·22-s − 0.585·23-s − 0.456·24-s + 0.475·25-s − 0.192·27-s + 0.0238·28-s − 0.389·29-s − 0.802·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3432282215\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3432282215\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 3.23T + 8T^{2} \) |
| 5 | \( 1 + 13.5T + 125T^{2} \) |
| 7 | \( 1 - 1.42T + 343T^{2} \) |
| 11 | \( 1 - 54.5T + 1.33e3T^{2} \) |
| 17 | \( 1 + 114.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 104.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 64.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 60.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 148.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 20.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 371.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 40.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 639.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 102.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 704.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 819.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 574.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 365.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 965.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 580.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 175.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 20.0T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.22e3T + 9.12e5T^{2} \) |
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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
−10.73724394345904612774833329158, −9.324951866702663274701802839940, −8.874593325156819979480941885679, −7.88747138156622577060442584839, −7.06867043611984573708590660077, −6.20535192126269998533209655263, −4.41725174790297334852867712752, −4.05047499629271399839897108139, −1.85350484237476979473755534740, −0.44352378685980362385502198572,
0.44352378685980362385502198572, 1.85350484237476979473755534740, 4.05047499629271399839897108139, 4.41725174790297334852867712752, 6.20535192126269998533209655263, 7.06867043611984573708590660077, 7.88747138156622577060442584839, 8.874593325156819979480941885679, 9.324951866702663274701802839940, 10.73724394345904612774833329158
