| L(s) = 1 | − 2.82·2-s − 0.801·3-s + 5.95·4-s − 2.47·5-s + 2.26·6-s − 0.795·7-s − 11.1·8-s − 2.35·9-s + 6.99·10-s − 2.73·11-s − 4.77·12-s + 2.24·14-s + 1.98·15-s + 19.5·16-s + 3.32·17-s + 6.64·18-s + 0.700·19-s − 14.7·20-s + 0.637·21-s + 7.72·22-s − 7.19·23-s + 8.94·24-s + 1.14·25-s + 4.29·27-s − 4.73·28-s − 0.681·29-s − 5.60·30-s + ⋯ |
| L(s) = 1 | − 1.99·2-s − 0.462·3-s + 2.97·4-s − 1.10·5-s + 0.923·6-s − 0.300·7-s − 3.94·8-s − 0.785·9-s + 2.21·10-s − 0.825·11-s − 1.37·12-s + 0.599·14-s + 0.513·15-s + 4.88·16-s + 0.807·17-s + 1.56·18-s + 0.160·19-s − 3.30·20-s + 0.139·21-s + 1.64·22-s − 1.49·23-s + 1.82·24-s + 0.229·25-s + 0.826·27-s − 0.895·28-s − 0.126·29-s − 1.02·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 2.82T + 2T^{2} \) |
| 3 | \( 1 + 0.801T + 3T^{2} \) |
| 5 | \( 1 + 2.47T + 5T^{2} \) |
| 7 | \( 1 + 0.795T + 7T^{2} \) |
| 11 | \( 1 + 2.73T + 11T^{2} \) |
| 17 | \( 1 - 3.32T + 17T^{2} \) |
| 19 | \( 1 - 0.700T + 19T^{2} \) |
| 23 | \( 1 + 7.19T + 23T^{2} \) |
| 29 | \( 1 + 0.681T + 29T^{2} \) |
| 37 | \( 1 - 7.81T + 37T^{2} \) |
| 41 | \( 1 + 2.36T + 41T^{2} \) |
| 43 | \( 1 - 3.63T + 43T^{2} \) |
| 47 | \( 1 - 2.34T + 47T^{2} \) |
| 53 | \( 1 - 3.27T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 4.87T + 61T^{2} \) |
| 67 | \( 1 - 9.54T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 - 8.22T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 - 0.671T + 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
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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
−7.955821763305713704783286793940, −7.60103547108935647437190155173, −6.60298375251291233466329689698, −6.02095729973037318062551102737, −5.27517472222301401049762771265, −3.75436971367065763558411253462, −2.98668368003501059938500120638, −2.13696297715745556974910723406, −0.76571872836381446200720213641, 0,
0.76571872836381446200720213641, 2.13696297715745556974910723406, 2.98668368003501059938500120638, 3.75436971367065763558411253462, 5.27517472222301401049762771265, 6.02095729973037318062551102737, 6.60298375251291233466329689698, 7.60103547108935647437190155173, 7.955821763305713704783286793940
