| L(s) = 1 | − 1.49·2-s + 3-s + 0.233·4-s − 1.69·5-s − 1.49·6-s + 0.268·7-s + 2.63·8-s + 9-s + 2.53·10-s + 0.913·11-s + 0.233·12-s + 6.63·13-s − 0.401·14-s − 1.69·15-s − 4.41·16-s + 17-s − 1.49·18-s + 0.913·19-s − 0.396·20-s + 0.268·21-s − 1.36·22-s + 4.49·23-s + 2.63·24-s − 2.12·25-s − 9.91·26-s + 27-s + 0.0627·28-s + ⋯ |
| L(s) = 1 | − 1.05·2-s + 0.577·3-s + 0.116·4-s − 0.758·5-s − 0.610·6-s + 0.101·7-s + 0.933·8-s + 0.333·9-s + 0.801·10-s + 0.275·11-s + 0.0674·12-s + 1.83·13-s − 0.107·14-s − 0.437·15-s − 1.10·16-s + 0.242·17-s − 0.352·18-s + 0.209·19-s − 0.0885·20-s + 0.0586·21-s − 0.291·22-s + 0.936·23-s + 0.538·24-s − 0.425·25-s − 1.94·26-s + 0.192·27-s + 0.0118·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.267318061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.267318061\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 + 1.49T + 2T^{2} \) |
| 5 | \( 1 + 1.69T + 5T^{2} \) |
| 7 | \( 1 - 0.268T + 7T^{2} \) |
| 11 | \( 1 - 0.913T + 11T^{2} \) |
| 13 | \( 1 - 6.63T + 13T^{2} \) |
| 19 | \( 1 - 0.913T + 19T^{2} \) |
| 23 | \( 1 - 4.49T + 23T^{2} \) |
| 29 | \( 1 - 5.79T + 29T^{2} \) |
| 31 | \( 1 + 6.13T + 31T^{2} \) |
| 37 | \( 1 - 0.597T + 37T^{2} \) |
| 41 | \( 1 - 6.73T + 41T^{2} \) |
| 43 | \( 1 - 6.30T + 43T^{2} \) |
| 47 | \( 1 - 8.20T + 47T^{2} \) |
| 53 | \( 1 - 7.64T + 53T^{2} \) |
| 59 | \( 1 + 13.4T + 59T^{2} \) |
| 61 | \( 1 - 0.904T + 61T^{2} \) |
| 67 | \( 1 + 5.93T + 67T^{2} \) |
| 71 | \( 1 - 3.32T + 71T^{2} \) |
| 73 | \( 1 + 0.542T + 73T^{2} \) |
| 83 | \( 1 + 3.99T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + 16.5T + 97T^{2} \) |
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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
−8.589938618817609329612301167200, −7.83223109991273542038944780713, −7.40613047445351973667591502108, −6.49682895678020327193931596717, −5.53627747310447031659574904855, −4.34954672164139697155877567391, −3.88653947088449973036623511462, −2.94173431887232394066743752807, −1.58541151613233807306223797749, −0.809099136873830277606804673562,
0.809099136873830277606804673562, 1.58541151613233807306223797749, 2.94173431887232394066743752807, 3.88653947088449973036623511462, 4.34954672164139697155877567391, 5.53627747310447031659574904855, 6.49682895678020327193931596717, 7.40613047445351973667591502108, 7.83223109991273542038944780713, 8.589938618817609329612301167200
