| L(s) = 1 | + 0.278·3-s − 0.533·5-s − 1.93·7-s − 2.92·9-s + 3.65·11-s + 5.69·13-s − 0.148·15-s + 6.46·17-s + 4.37·19-s − 0.540·21-s + 8.89·23-s − 4.71·25-s − 1.65·27-s − 0.939·29-s − 6.67·31-s + 1.01·33-s + 1.03·35-s + 0.254·37-s + 1.58·39-s + 1.94·41-s − 1.83·43-s + 1.55·45-s + 5.30·47-s − 3.24·49-s + 1.80·51-s + 1.23·53-s − 1.94·55-s + ⋯ |
| L(s) = 1 | + 0.160·3-s − 0.238·5-s − 0.732·7-s − 0.974·9-s + 1.10·11-s + 1.58·13-s − 0.0383·15-s + 1.56·17-s + 1.00·19-s − 0.117·21-s + 1.85·23-s − 0.943·25-s − 0.317·27-s − 0.174·29-s − 1.19·31-s + 0.177·33-s + 0.174·35-s + 0.0419·37-s + 0.254·39-s + 0.304·41-s − 0.279·43-s + 0.232·45-s + 0.774·47-s − 0.463·49-s + 0.252·51-s + 0.170·53-s − 0.262·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.209872988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.209872988\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 - 0.278T + 3T^{2} \) |
| 5 | \( 1 + 0.533T + 5T^{2} \) |
| 7 | \( 1 + 1.93T + 7T^{2} \) |
| 11 | \( 1 - 3.65T + 11T^{2} \) |
| 13 | \( 1 - 5.69T + 13T^{2} \) |
| 17 | \( 1 - 6.46T + 17T^{2} \) |
| 19 | \( 1 - 4.37T + 19T^{2} \) |
| 23 | \( 1 - 8.89T + 23T^{2} \) |
| 29 | \( 1 + 0.939T + 29T^{2} \) |
| 31 | \( 1 + 6.67T + 31T^{2} \) |
| 37 | \( 1 - 0.254T + 37T^{2} \) |
| 41 | \( 1 - 1.94T + 41T^{2} \) |
| 43 | \( 1 + 1.83T + 43T^{2} \) |
| 47 | \( 1 - 5.30T + 47T^{2} \) |
| 53 | \( 1 - 1.23T + 53T^{2} \) |
| 59 | \( 1 - 1.09T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 9.49T + 71T^{2} \) |
| 73 | \( 1 + 0.549T + 73T^{2} \) |
| 79 | \( 1 - 1.46T + 79T^{2} \) |
| 83 | \( 1 + 2.74T + 83T^{2} \) |
| 89 | \( 1 - 8.78T + 89T^{2} \) |
| 97 | \( 1 + 8.40T + 97T^{2} \) |
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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
−7.78181772475303556163236477275, −7.21281186672235439964901884312, −6.32520205798526860949571452804, −5.82980925980483639579780198197, −5.22895755272000685221134117040, −4.00778230215104012186613612642, −3.32211454974843398562821032056, −3.11748573073218242883307670450, −1.58862393032499314271358574308, −0.78036431386359459043495303434,
0.78036431386359459043495303434, 1.58862393032499314271358574308, 3.11748573073218242883307670450, 3.32211454974843398562821032056, 4.00778230215104012186613612642, 5.22895755272000685221134117040, 5.82980925980483639579780198197, 6.32520205798526860949571452804, 7.21281186672235439964901884312, 7.78181772475303556163236477275
