L(s) = 1 | + 2-s + 2.90·3-s + 4-s + 2.46·5-s + 2.90·6-s + 0.871·7-s + 8-s + 5.45·9-s + 2.46·10-s + 4.42·11-s + 2.90·12-s − 6.00·13-s + 0.871·14-s + 7.18·15-s + 16-s − 0.755·17-s + 5.45·18-s − 2.18·19-s + 2.46·20-s + 2.53·21-s + 4.42·22-s − 3.76·23-s + 2.90·24-s + 1.09·25-s − 6.00·26-s + 7.14·27-s + 0.871·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.67·3-s + 0.5·4-s + 1.10·5-s + 1.18·6-s + 0.329·7-s + 0.353·8-s + 1.81·9-s + 0.780·10-s + 1.33·11-s + 0.839·12-s − 1.66·13-s + 0.232·14-s + 1.85·15-s + 0.250·16-s − 0.183·17-s + 1.28·18-s − 0.500·19-s + 0.552·20-s + 0.553·21-s + 0.942·22-s − 0.785·23-s + 0.593·24-s + 0.219·25-s − 1.17·26-s + 1.37·27-s + 0.164·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.112045822\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.112045822\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 2011 | \( 1 + T \) |
good | 3 | \( 1 - 2.90T + 3T^{2} \) |
| 5 | \( 1 - 2.46T + 5T^{2} \) |
| 7 | \( 1 - 0.871T + 7T^{2} \) |
| 11 | \( 1 - 4.42T + 11T^{2} \) |
| 13 | \( 1 + 6.00T + 13T^{2} \) |
| 17 | \( 1 + 0.755T + 17T^{2} \) |
| 19 | \( 1 + 2.18T + 19T^{2} \) |
| 23 | \( 1 + 3.76T + 23T^{2} \) |
| 29 | \( 1 + 7.80T + 29T^{2} \) |
| 31 | \( 1 - 1.67T + 31T^{2} \) |
| 37 | \( 1 - 2.75T + 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 + 3.33T + 43T^{2} \) |
| 47 | \( 1 - 8.80T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 - 9.07T + 59T^{2} \) |
| 61 | \( 1 + 1.40T + 61T^{2} \) |
| 67 | \( 1 - 4.06T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 + 9.72T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 + 1.35T + 83T^{2} \) |
| 89 | \( 1 + 9.23T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−8.571735440393639971708207082040, −7.52951649384554414040248578284, −7.18640740921897122008712714888, −6.19134724196344413822138279290, −5.46248777208273475769242716986, −4.29902177153614209988362575941, −3.98432300641992594411186235645, −2.73899289700243599231392219746, −2.24200575931739849892458048528, −1.55423189609727758173073634713,
1.55423189609727758173073634713, 2.24200575931739849892458048528, 2.73899289700243599231392219746, 3.98432300641992594411186235645, 4.29902177153614209988362575941, 5.46248777208273475769242716986, 6.19134724196344413822138279290, 7.18640740921897122008712714888, 7.52951649384554414040248578284, 8.571735440393639971708207082040