L(s) = 1 | − 2.74·2-s − 1.76·3-s + 5.53·4-s − 2.91·5-s + 4.83·6-s + 1.39·7-s − 9.70·8-s + 0.0985·9-s + 8.01·10-s − 11-s − 9.74·12-s + 4.17·13-s − 3.82·14-s + 5.13·15-s + 15.5·16-s − 4.04·17-s − 0.270·18-s + 4.18·19-s − 16.1·20-s − 2.45·21-s + 2.74·22-s + 0.233·23-s + 17.0·24-s + 3.52·25-s − 11.4·26-s + 5.10·27-s + 7.71·28-s + ⋯ |
L(s) = 1 | − 1.94·2-s − 1.01·3-s + 2.76·4-s − 1.30·5-s + 1.97·6-s + 0.526·7-s − 3.43·8-s + 0.0328·9-s + 2.53·10-s − 0.301·11-s − 2.81·12-s + 1.15·13-s − 1.02·14-s + 1.32·15-s + 3.89·16-s − 0.980·17-s − 0.0637·18-s + 0.960·19-s − 3.61·20-s − 0.535·21-s + 0.585·22-s + 0.0486·23-s + 3.48·24-s + 0.704·25-s − 2.24·26-s + 0.982·27-s + 1.45·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3256114606\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3256114606\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + T \) |
| 547 | \( 1 - T \) |
good | 2 | \( 1 + 2.74T + 2T^{2} \) |
| 3 | \( 1 + 1.76T + 3T^{2} \) |
| 5 | \( 1 + 2.91T + 5T^{2} \) |
| 7 | \( 1 - 1.39T + 7T^{2} \) |
| 13 | \( 1 - 4.17T + 13T^{2} \) |
| 17 | \( 1 + 4.04T + 17T^{2} \) |
| 19 | \( 1 - 4.18T + 19T^{2} \) |
| 23 | \( 1 - 0.233T + 23T^{2} \) |
| 29 | \( 1 - 6.60T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 0.0900T + 37T^{2} \) |
| 41 | \( 1 - 5.18T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 2.66T + 53T^{2} \) |
| 59 | \( 1 - 7.50T + 59T^{2} \) |
| 61 | \( 1 + 2.25T + 61T^{2} \) |
| 67 | \( 1 + 8.19T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + 7.55T + 79T^{2} \) |
| 83 | \( 1 + 0.344T + 83T^{2} \) |
| 89 | \( 1 - 0.821T + 89T^{2} \) |
| 97 | \( 1 - 1.35T + 97T^{2} \) |
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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
−8.140109626556687835148096610863, −7.45253334173064367088395031737, −7.06627709013713050867418048971, −6.11865127587552019609933152781, −5.61486011564430438344275541700, −4.44764589455686557502464162685, −3.43509176596278431327856931497, −2.45883509934017683918072540727, −1.22153833549435245779175357059, −0.48911580467925967789562357251,
0.48911580467925967789562357251, 1.22153833549435245779175357059, 2.45883509934017683918072540727, 3.43509176596278431327856931497, 4.44764589455686557502464162685, 5.61486011564430438344275541700, 6.11865127587552019609933152781, 7.06627709013713050867418048971, 7.45253334173064367088395031737, 8.140109626556687835148096610863