L(s) = 1 | + 2.01·2-s − 2.84·3-s + 2.05·4-s − 1.15·5-s − 5.73·6-s − 7-s + 0.109·8-s + 5.11·9-s − 2.31·10-s − 6.44·11-s − 5.85·12-s + 4.28·13-s − 2.01·14-s + 3.27·15-s − 3.88·16-s − 5.39·17-s + 10.3·18-s + 3.71·19-s − 2.36·20-s + 2.84·21-s − 12.9·22-s + 4.71·23-s − 0.312·24-s − 3.67·25-s + 8.62·26-s − 6.02·27-s − 2.05·28-s + ⋯ |
L(s) = 1 | + 1.42·2-s − 1.64·3-s + 1.02·4-s − 0.514·5-s − 2.34·6-s − 0.377·7-s + 0.0387·8-s + 1.70·9-s − 0.732·10-s − 1.94·11-s − 1.68·12-s + 1.18·13-s − 0.538·14-s + 0.845·15-s − 0.972·16-s − 1.30·17-s + 2.42·18-s + 0.852·19-s − 0.528·20-s + 0.621·21-s − 2.76·22-s + 0.984·23-s − 0.0637·24-s − 0.735·25-s + 1.69·26-s − 1.16·27-s − 0.388·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.162290267\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.162290267\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + T \) |
| 431 | \( 1 - T \) |
good | 2 | \( 1 - 2.01T + 2T^{2} \) |
| 3 | \( 1 + 2.84T + 3T^{2} \) |
| 5 | \( 1 + 1.15T + 5T^{2} \) |
| 11 | \( 1 + 6.44T + 11T^{2} \) |
| 13 | \( 1 - 4.28T + 13T^{2} \) |
| 17 | \( 1 + 5.39T + 17T^{2} \) |
| 19 | \( 1 - 3.71T + 19T^{2} \) |
| 23 | \( 1 - 4.71T + 23T^{2} \) |
| 29 | \( 1 + 6.29T + 29T^{2} \) |
| 31 | \( 1 - 5.95T + 31T^{2} \) |
| 37 | \( 1 + 2.57T + 37T^{2} \) |
| 41 | \( 1 - 1.45T + 41T^{2} \) |
| 43 | \( 1 + 0.366T + 43T^{2} \) |
| 47 | \( 1 - 5.93T + 47T^{2} \) |
| 53 | \( 1 - 5.18T + 53T^{2} \) |
| 59 | \( 1 - 1.02T + 59T^{2} \) |
| 61 | \( 1 - 1.51T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 6.19T + 71T^{2} \) |
| 73 | \( 1 - 2.82T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 0.0872T + 83T^{2} \) |
| 89 | \( 1 - 5.89T + 89T^{2} \) |
| 97 | \( 1 + 1.67T + 97T^{2} \) |
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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
−8.631586852819720672207681000902, −7.59148456588315362913023798814, −6.83394947203620621034915271926, −6.15954055224519630302782453732, −5.45587709527734712831174565360, −5.07480735567713022087933909995, −4.23269866207200012827854798921, −3.43880263888330240616876780228, −2.36011667282416646278356944732, −0.54826880121649188609797945150,
0.54826880121649188609797945150, 2.36011667282416646278356944732, 3.43880263888330240616876780228, 4.23269866207200012827854798921, 5.07480735567713022087933909995, 5.45587709527734712831174565360, 6.15954055224519630302782453732, 6.83394947203620621034915271926, 7.59148456588315362913023798814, 8.631586852819720672207681000902