| L(s) = 1 | − 2-s − 1.70·3-s + 4-s − 2.14·5-s + 1.70·6-s + 0.462·7-s − 8-s − 0.0885·9-s + 2.14·10-s − 3.41·11-s − 1.70·12-s − 6.04·13-s − 0.462·14-s + 3.66·15-s + 16-s − 2.77·17-s + 0.0885·18-s + 4.36·19-s − 2.14·20-s − 0.788·21-s + 3.41·22-s + 23-s + 1.70·24-s − 0.384·25-s + 6.04·26-s + 5.26·27-s + 0.462·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.985·3-s + 0.5·4-s − 0.960·5-s + 0.696·6-s + 0.174·7-s − 0.353·8-s − 0.0295·9-s + 0.679·10-s − 1.02·11-s − 0.492·12-s − 1.67·13-s − 0.123·14-s + 0.946·15-s + 0.250·16-s − 0.672·17-s + 0.0208·18-s + 1.00·19-s − 0.480·20-s − 0.172·21-s + 0.727·22-s + 0.208·23-s + 0.348·24-s − 0.0768·25-s + 1.18·26-s + 1.01·27-s + 0.0873·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 + 1.70T + 3T^{2} \) |
| 5 | \( 1 + 2.14T + 5T^{2} \) |
| 7 | \( 1 - 0.462T + 7T^{2} \) |
| 11 | \( 1 + 3.41T + 11T^{2} \) |
| 13 | \( 1 + 6.04T + 13T^{2} \) |
| 17 | \( 1 + 2.77T + 17T^{2} \) |
| 19 | \( 1 - 4.36T + 19T^{2} \) |
| 29 | \( 1 - 6.57T + 29T^{2} \) |
| 31 | \( 1 - 5.44T + 31T^{2} \) |
| 37 | \( 1 + 6.84T + 37T^{2} \) |
| 41 | \( 1 - 1.53T + 41T^{2} \) |
| 43 | \( 1 + 2.73T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 + 5.46T + 59T^{2} \) |
| 61 | \( 1 + 6.36T + 61T^{2} \) |
| 67 | \( 1 - 8.91T + 67T^{2} \) |
| 71 | \( 1 - 4.61T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 - 5.62T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 + 5.27T + 89T^{2} \) |
| 97 | \( 1 + 0.514T + 97T^{2} \) |
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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
−7.85900553918447045691094219686, −7.03171846831486156518479507858, −6.57050688383065182203390663704, −5.35541518837011565659180763363, −5.10881068187339663960048937152, −4.21153209855504715739986629653, −2.97980825996065986081788185743, −2.35328642973828310463901600718, −0.795886524362455280404955689776, 0,
0.795886524362455280404955689776, 2.35328642973828310463901600718, 2.97980825996065986081788185743, 4.21153209855504715739986629653, 5.10881068187339663960048937152, 5.35541518837011565659180763363, 6.57050688383065182203390663704, 7.03171846831486156518479507858, 7.85900553918447045691094219686
