L(s) = 1 | − 0.775·2-s + 3-s − 1.39·4-s − 3.03·5-s − 0.775·6-s + 7-s + 2.63·8-s + 9-s + 2.35·10-s + 6.14·11-s − 1.39·12-s − 0.775·14-s − 3.03·15-s + 0.750·16-s + 6.44·17-s − 0.775·18-s + 4.32·19-s + 4.24·20-s + 21-s − 4.76·22-s − 8.92·23-s + 2.63·24-s + 4.20·25-s + 27-s − 1.39·28-s − 2.44·29-s + 2.35·30-s + ⋯ |
L(s) = 1 | − 0.548·2-s + 0.577·3-s − 0.699·4-s − 1.35·5-s − 0.316·6-s + 0.377·7-s + 0.932·8-s + 0.333·9-s + 0.744·10-s + 1.85·11-s − 0.403·12-s − 0.207·14-s − 0.783·15-s + 0.187·16-s + 1.56·17-s − 0.182·18-s + 0.991·19-s + 0.948·20-s + 0.218·21-s − 1.01·22-s − 1.86·23-s + 0.538·24-s + 0.841·25-s + 0.192·27-s − 0.264·28-s − 0.454·29-s + 0.429·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.352263674\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.352263674\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 0.775T + 2T^{2} \) |
| 5 | \( 1 + 3.03T + 5T^{2} \) |
| 11 | \( 1 - 6.14T + 11T^{2} \) |
| 17 | \( 1 - 6.44T + 17T^{2} \) |
| 19 | \( 1 - 4.32T + 19T^{2} \) |
| 23 | \( 1 + 8.92T + 23T^{2} \) |
| 29 | \( 1 + 2.44T + 29T^{2} \) |
| 31 | \( 1 - 3.00T + 31T^{2} \) |
| 37 | \( 1 - 2.38T + 37T^{2} \) |
| 41 | \( 1 + 0.945T + 41T^{2} \) |
| 43 | \( 1 + 5.28T + 43T^{2} \) |
| 47 | \( 1 + 0.212T + 47T^{2} \) |
| 53 | \( 1 - 4.03T + 53T^{2} \) |
| 59 | \( 1 - 1.72T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 - 6.35T + 67T^{2} \) |
| 71 | \( 1 - 7.06T + 71T^{2} \) |
| 73 | \( 1 + 1.75T + 73T^{2} \) |
| 79 | \( 1 - 17.3T + 79T^{2} \) |
| 83 | \( 1 - 4.99T + 83T^{2} \) |
| 89 | \( 1 - 3.56T + 89T^{2} \) |
| 97 | \( 1 - 1.57T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.411408952927072955278455818654, −7.903980498795821600533128739841, −7.56696734532001033016003972713, −6.56483743409865051513273433870, −5.44803554346066571775040824393, −4.43349078949349770355500932674, −3.83224606108091787498368238543, −3.39666760637699563683650262955, −1.66227577356014225012625103072, −0.791283921873866277957190995171,
0.791283921873866277957190995171, 1.66227577356014225012625103072, 3.39666760637699563683650262955, 3.83224606108091787498368238543, 4.43349078949349770355500932674, 5.44803554346066571775040824393, 6.56483743409865051513273433870, 7.56696734532001033016003972713, 7.903980498795821600533128739841, 8.411408952927072955278455818654