L(s) = 1 | + 2-s + 4-s − 4.60·7-s + 8-s + 2.69·11-s + 1.69·13-s − 4.60·14-s + 16-s − 2.60·17-s + 4.30·19-s + 2.69·22-s − 4·23-s − 5·25-s + 1.69·26-s − 4.60·28-s + 5.30·29-s + 6.60·31-s + 32-s − 2.60·34-s + 4.60·37-s + 4.30·38-s − 4·41-s + 3.90·43-s + 2.69·44-s − 4·46-s + 0.697·47-s + 14.2·49-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.74·7-s + 0.353·8-s + 0.813·11-s + 0.470·13-s − 1.23·14-s + 0.250·16-s − 0.631·17-s + 0.987·19-s + 0.575·22-s − 0.834·23-s − 25-s + 0.332·26-s − 0.870·28-s + 0.984·29-s + 1.18·31-s + 0.176·32-s − 0.446·34-s + 0.757·37-s + 0.698·38-s − 0.624·41-s + 0.596·43-s + 0.406·44-s − 0.589·46-s + 0.101·47-s + 2.03·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.535397100\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.535397100\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 223 | \( 1 + T \) |
good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 4.60T + 7T^{2} \) |
| 11 | \( 1 - 2.69T + 11T^{2} \) |
| 13 | \( 1 - 1.69T + 13T^{2} \) |
| 17 | \( 1 + 2.60T + 17T^{2} \) |
| 19 | \( 1 - 4.30T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 5.30T + 29T^{2} \) |
| 31 | \( 1 - 6.60T + 31T^{2} \) |
| 37 | \( 1 - 4.60T + 37T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 - 3.90T + 43T^{2} \) |
| 47 | \( 1 - 0.697T + 47T^{2} \) |
| 53 | \( 1 - 6.90T + 53T^{2} \) |
| 59 | \( 1 - 0.302T + 59T^{2} \) |
| 61 | \( 1 + 2.69T + 61T^{2} \) |
| 67 | \( 1 - 4.60T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 + 8.90T + 73T^{2} \) |
| 79 | \( 1 - 8.51T + 79T^{2} \) |
| 83 | \( 1 + 5.21T + 83T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 + 0.788T + 97T^{2} \) |
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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
−8.452512580828103690275563107163, −7.52128082438125781523221355795, −6.65292165646271728631785750404, −6.29405210868210541701229249654, −5.66554143417133303229295296261, −4.49819863725997226427627388022, −3.78655078297298587025780136077, −3.16256737062436172329747616053, −2.24763094074381032511929667268, −0.813117710041396841667166824850,
0.813117710041396841667166824850, 2.24763094074381032511929667268, 3.16256737062436172329747616053, 3.78655078297298587025780136077, 4.49819863725997226427627388022, 5.66554143417133303229295296261, 6.29405210868210541701229249654, 6.65292165646271728631785750404, 7.52128082438125781523221355795, 8.452512580828103690275563107163