| L(s) = 1 | − 2-s + 4-s − 1.96·5-s − 4.72·7-s − 8-s + 1.96·10-s − 2.36·11-s − 0.742·13-s + 4.72·14-s + 16-s + 1.40·17-s − 2.69·19-s − 1.96·20-s + 2.36·22-s + 3.63·23-s − 1.13·25-s + 0.742·26-s − 4.72·28-s − 0.980·29-s + 3.58·31-s − 32-s − 1.40·34-s + 9.28·35-s + 10.6·37-s + 2.69·38-s + 1.96·40-s + 6.88·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.879·5-s − 1.78·7-s − 0.353·8-s + 0.621·10-s − 0.713·11-s − 0.205·13-s + 1.26·14-s + 0.250·16-s + 0.339·17-s − 0.617·19-s − 0.439·20-s + 0.504·22-s + 0.758·23-s − 0.226·25-s + 0.145·26-s − 0.892·28-s − 0.182·29-s + 0.644·31-s − 0.176·32-s − 0.240·34-s + 1.56·35-s + 1.74·37-s + 0.436·38-s + 0.310·40-s + 1.07·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 5 | \( 1 + 1.96T + 5T^{2} \) |
| 7 | \( 1 + 4.72T + 7T^{2} \) |
| 11 | \( 1 + 2.36T + 11T^{2} \) |
| 13 | \( 1 + 0.742T + 13T^{2} \) |
| 17 | \( 1 - 1.40T + 17T^{2} \) |
| 19 | \( 1 + 2.69T + 19T^{2} \) |
| 23 | \( 1 - 3.63T + 23T^{2} \) |
| 29 | \( 1 + 0.980T + 29T^{2} \) |
| 31 | \( 1 - 3.58T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 - 6.88T + 41T^{2} \) |
| 43 | \( 1 + 1.80T + 43T^{2} \) |
| 47 | \( 1 - 2.73T + 47T^{2} \) |
| 53 | \( 1 + 1.45T + 53T^{2} \) |
| 59 | \( 1 - 2.49T + 59T^{2} \) |
| 61 | \( 1 + 7.04T + 61T^{2} \) |
| 67 | \( 1 + 4.95T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 + 7.71T + 73T^{2} \) |
| 79 | \( 1 - 2.59T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 + 1.95T + 89T^{2} \) |
| 97 | \( 1 - 0.529T + 97T^{2} \) |
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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
−7.63230756316571370614626446840, −6.87909691822810180210170485953, −6.28045711343648264640813032621, −5.63763685132811861509886358644, −4.52410521678097111470427087825, −3.74053654325833117593427341400, −2.98417109403947272724540474785, −2.41940742307949299457775225947, −0.835655961346648460693801053990, 0,
0.835655961346648460693801053990, 2.41940742307949299457775225947, 2.98417109403947272724540474785, 3.74053654325833117593427341400, 4.52410521678097111470427087825, 5.63763685132811861509886358644, 6.28045711343648264640813032621, 6.87909691822810180210170485953, 7.63230756316571370614626446840
