| L(s) = 1 | − 0.607·2-s + 2.06·3-s − 1.63·4-s + 2.69·5-s − 1.25·6-s − 3.19·7-s + 2.20·8-s + 1.28·9-s − 1.63·10-s + 4.87·11-s − 3.37·12-s + 1.84·13-s + 1.94·14-s + 5.57·15-s + 1.92·16-s − 17-s − 0.777·18-s − 0.104·19-s − 4.39·20-s − 6.61·21-s − 2.96·22-s + 0.149·23-s + 4.56·24-s + 2.25·25-s − 1.12·26-s − 3.55·27-s + 5.21·28-s + ⋯ |
| L(s) = 1 | − 0.429·2-s + 1.19·3-s − 0.815·4-s + 1.20·5-s − 0.513·6-s − 1.20·7-s + 0.779·8-s + 0.426·9-s − 0.517·10-s + 1.47·11-s − 0.974·12-s + 0.512·13-s + 0.519·14-s + 1.43·15-s + 0.480·16-s − 0.242·17-s − 0.183·18-s − 0.0239·19-s − 0.982·20-s − 1.44·21-s − 0.631·22-s + 0.0312·23-s + 0.931·24-s + 0.451·25-s − 0.219·26-s − 0.684·27-s + 0.985·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.895672984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.895672984\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 2 | \( 1 + 0.607T + 2T^{2} \) |
| 3 | \( 1 - 2.06T + 3T^{2} \) |
| 5 | \( 1 - 2.69T + 5T^{2} \) |
| 7 | \( 1 + 3.19T + 7T^{2} \) |
| 11 | \( 1 - 4.87T + 11T^{2} \) |
| 13 | \( 1 - 1.84T + 13T^{2} \) |
| 19 | \( 1 + 0.104T + 19T^{2} \) |
| 23 | \( 1 - 0.149T + 23T^{2} \) |
| 29 | \( 1 - 2.79T + 29T^{2} \) |
| 31 | \( 1 - 8.11T + 31T^{2} \) |
| 37 | \( 1 - 11.5T + 37T^{2} \) |
| 41 | \( 1 + 1.08T + 41T^{2} \) |
| 43 | \( 1 - 7.77T + 43T^{2} \) |
| 47 | \( 1 + 5.58T + 47T^{2} \) |
| 53 | \( 1 - 3.52T + 53T^{2} \) |
| 61 | \( 1 + 3.10T + 61T^{2} \) |
| 67 | \( 1 - 4.11T + 67T^{2} \) |
| 71 | \( 1 + 9.76T + 71T^{2} \) |
| 73 | \( 1 + 9.19T + 73T^{2} \) |
| 79 | \( 1 + 2.57T + 79T^{2} \) |
| 83 | \( 1 - 9.67T + 83T^{2} \) |
| 89 | \( 1 - 4.78T + 89T^{2} \) |
| 97 | \( 1 - 0.0244T + 97T^{2} \) |
| 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
−9.756605664145179201687731921076, −9.087149439647971328952865887926, −8.760064076103538799752543072094, −7.72008863025386420502500951556, −6.47667669615450411886830344783, −5.96587722298423324286687653145, −4.44488477278191038864673434655, −3.56187079918161661465105277202, −2.54404357047808312788949904127, −1.19963442862433363549706513225,
1.19963442862433363549706513225, 2.54404357047808312788949904127, 3.56187079918161661465105277202, 4.44488477278191038864673434655, 5.96587722298423324286687653145, 6.47667669615450411886830344783, 7.72008863025386420502500951556, 8.760064076103538799752543072094, 9.087149439647971328952865887926, 9.756605664145179201687731921076
