| L(s) = 1 | + 0.300·2-s − 1.90·4-s + 3.45·5-s − 1.15·7-s − 1.17·8-s + 1.03·10-s − 6.20·11-s − 0.943·13-s − 0.346·14-s + 3.46·16-s + 7.87·17-s − 8.23·19-s − 6.60·20-s − 1.86·22-s + 23-s + 6.94·25-s − 0.283·26-s + 2.20·28-s + 29-s − 3.76·31-s + 3.38·32-s + 2.36·34-s − 3.98·35-s + 7.78·37-s − 2.47·38-s − 4.05·40-s + 0.655·41-s + ⋯ |
| L(s) = 1 | + 0.212·2-s − 0.954·4-s + 1.54·5-s − 0.435·7-s − 0.414·8-s + 0.328·10-s − 1.87·11-s − 0.261·13-s − 0.0925·14-s + 0.866·16-s + 1.91·17-s − 1.88·19-s − 1.47·20-s − 0.397·22-s + 0.208·23-s + 1.38·25-s − 0.0555·26-s + 0.416·28-s + 0.185·29-s − 0.675·31-s + 0.599·32-s + 0.405·34-s − 0.673·35-s + 1.27·37-s − 0.401·38-s − 0.641·40-s + 0.102·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.677443490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.677443490\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 0.300T + 2T^{2} \) |
| 5 | \( 1 - 3.45T + 5T^{2} \) |
| 7 | \( 1 + 1.15T + 7T^{2} \) |
| 11 | \( 1 + 6.20T + 11T^{2} \) |
| 13 | \( 1 + 0.943T + 13T^{2} \) |
| 17 | \( 1 - 7.87T + 17T^{2} \) |
| 19 | \( 1 + 8.23T + 19T^{2} \) |
| 31 | \( 1 + 3.76T + 31T^{2} \) |
| 37 | \( 1 - 7.78T + 37T^{2} \) |
| 41 | \( 1 - 0.655T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 + 6.17T + 53T^{2} \) |
| 59 | \( 1 - 2.25T + 59T^{2} \) |
| 61 | \( 1 - 3.80T + 61T^{2} \) |
| 67 | \( 1 - 4.88T + 67T^{2} \) |
| 71 | \( 1 + 2.81T + 71T^{2} \) |
| 73 | \( 1 - 8.34T + 73T^{2} \) |
| 79 | \( 1 - 17.5T + 79T^{2} \) |
| 83 | \( 1 + 4.47T + 83T^{2} \) |
| 89 | \( 1 - 0.0918T + 89T^{2} \) |
| 97 | \( 1 - 2.85T + 97T^{2} \) |
| 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
−8.079163015146466947128619122500, −7.54962868309083399469797269208, −6.24963723045442808460077788777, −5.94423380006450771794735368500, −5.16605039797032286591298275634, −4.76785445846015422252653040543, −3.55685130504131304653532370182, −2.76009138591294645021490694571, −2.00983227029285364301804228204, −0.64509624445262566821255078282,
0.64509624445262566821255078282, 2.00983227029285364301804228204, 2.76009138591294645021490694571, 3.55685130504131304653532370182, 4.76785445846015422252653040543, 5.16605039797032286591298275634, 5.94423380006450771794735368500, 6.24963723045442808460077788777, 7.54962868309083399469797269208, 8.079163015146466947128619122500
