L(s) = 1 | − 2-s + 2.30·3-s + 4-s + 5-s − 2.30·6-s − 5.12·7-s − 8-s + 2.32·9-s − 10-s − 1.82·11-s + 2.30·12-s + 2.86·13-s + 5.12·14-s + 2.30·15-s + 16-s + 5.18·17-s − 2.32·18-s − 1.72·19-s + 20-s − 11.8·21-s + 1.82·22-s − 0.231·23-s − 2.30·24-s + 25-s − 2.86·26-s − 1.55·27-s − 5.12·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.33·3-s + 0.5·4-s + 0.447·5-s − 0.942·6-s − 1.93·7-s − 0.353·8-s + 0.775·9-s − 0.316·10-s − 0.550·11-s + 0.666·12-s + 0.795·13-s + 1.36·14-s + 0.595·15-s + 0.250·16-s + 1.25·17-s − 0.548·18-s − 0.395·19-s + 0.223·20-s − 2.58·21-s + 0.389·22-s − 0.0483·23-s − 0.471·24-s + 0.200·25-s − 0.562·26-s − 0.299·27-s − 0.968·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 - 2.30T + 3T^{2} \) |
| 7 | \( 1 + 5.12T + 7T^{2} \) |
| 11 | \( 1 + 1.82T + 11T^{2} \) |
| 13 | \( 1 - 2.86T + 13T^{2} \) |
| 17 | \( 1 - 5.18T + 17T^{2} \) |
| 19 | \( 1 + 1.72T + 19T^{2} \) |
| 23 | \( 1 + 0.231T + 23T^{2} \) |
| 29 | \( 1 - 0.413T + 29T^{2} \) |
| 31 | \( 1 + 5.80T + 31T^{2} \) |
| 37 | \( 1 + 0.773T + 37T^{2} \) |
| 41 | \( 1 + 8.47T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + 13.3T + 47T^{2} \) |
| 53 | \( 1 - 3.57T + 53T^{2} \) |
| 59 | \( 1 + 8.35T + 59T^{2} \) |
| 61 | \( 1 - 7.09T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 + 7.02T + 71T^{2} \) |
| 73 | \( 1 + 7.84T + 73T^{2} \) |
| 79 | \( 1 - 8.27T + 79T^{2} \) |
| 83 | \( 1 - 2.73T + 83T^{2} \) |
| 89 | \( 1 - 1.59T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−8.319021853054874206447632820104, −7.50047079412507067232580469808, −6.72757513278983424528819927734, −6.13208626723513572867987527079, −5.26614695887212159545415139238, −3.54717049887883054460871976840, −3.40905533675463278514888034766, −2.55975168377336365124557300766, −1.56026130152732342930161177109, 0,
1.56026130152732342930161177109, 2.55975168377336365124557300766, 3.40905533675463278514888034766, 3.54717049887883054460871976840, 5.26614695887212159545415139238, 6.13208626723513572867987527079, 6.72757513278983424528819927734, 7.50047079412507067232580469808, 8.319021853054874206447632820104