| L(s) = 1 | − 1.57·2-s + 0.410·3-s + 0.479·4-s + 0.496·5-s − 0.646·6-s + 3.25·7-s + 2.39·8-s − 2.83·9-s − 0.781·10-s + 0.619·11-s + 0.196·12-s − 13-s − 5.12·14-s + 0.203·15-s − 4.72·16-s + 1.98·17-s + 4.45·18-s + 7.19·19-s + 0.237·20-s + 1.33·21-s − 0.976·22-s + 5.66·23-s + 0.982·24-s − 4.75·25-s + 1.57·26-s − 2.39·27-s + 1.56·28-s + ⋯ |
| L(s) = 1 | − 1.11·2-s + 0.237·3-s + 0.239·4-s + 0.221·5-s − 0.263·6-s + 1.22·7-s + 0.846·8-s − 0.943·9-s − 0.246·10-s + 0.186·11-s + 0.0568·12-s − 0.277·13-s − 1.36·14-s + 0.0525·15-s − 1.18·16-s + 0.481·17-s + 1.05·18-s + 1.64·19-s + 0.0531·20-s + 0.291·21-s − 0.208·22-s + 1.18·23-s + 0.200·24-s − 0.950·25-s + 0.308·26-s − 0.460·27-s + 0.294·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.058732740\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.058732740\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 + 1.57T + 2T^{2} \) |
| 3 | \( 1 - 0.410T + 3T^{2} \) |
| 5 | \( 1 - 0.496T + 5T^{2} \) |
| 7 | \( 1 - 3.25T + 7T^{2} \) |
| 11 | \( 1 - 0.619T + 11T^{2} \) |
| 17 | \( 1 - 1.98T + 17T^{2} \) |
| 19 | \( 1 - 7.19T + 19T^{2} \) |
| 23 | \( 1 - 5.66T + 23T^{2} \) |
| 29 | \( 1 + 0.707T + 29T^{2} \) |
| 31 | \( 1 - 3.09T + 31T^{2} \) |
| 37 | \( 1 + 7.09T + 37T^{2} \) |
| 41 | \( 1 + 0.437T + 41T^{2} \) |
| 43 | \( 1 + 4.60T + 43T^{2} \) |
| 47 | \( 1 - 6.23T + 47T^{2} \) |
| 53 | \( 1 + 3.89T + 53T^{2} \) |
| 59 | \( 1 - 6.74T + 59T^{2} \) |
| 61 | \( 1 - 8.96T + 61T^{2} \) |
| 67 | \( 1 - 14.6T + 67T^{2} \) |
| 71 | \( 1 - 8.90T + 71T^{2} \) |
| 73 | \( 1 + 8.85T + 73T^{2} \) |
| 83 | \( 1 - 8.48T + 83T^{2} \) |
| 89 | \( 1 - 4.58T + 89T^{2} \) |
| 97 | \( 1 - 9.87T + 97T^{2} \) |
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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
−9.734425766917984250060964193526, −9.095031377223540563897597524867, −8.307674037560650816662481325966, −7.79020299558615224964083071733, −6.96055200522778391605066487538, −5.47104397174523521022188796603, −4.93427919857610062655563504853, −3.50646723632815774705630525423, −2.14005978323220387335287651129, −0.986915799996203682053867556639,
0.986915799996203682053867556639, 2.14005978323220387335287651129, 3.50646723632815774705630525423, 4.93427919857610062655563504853, 5.47104397174523521022188796603, 6.96055200522778391605066487538, 7.79020299558615224964083071733, 8.307674037560650816662481325966, 9.095031377223540563897597524867, 9.734425766917984250060964193526
