| L(s) = 1 | − 2.33·2-s − 1.57·3-s + 3.44·4-s + 2.09·5-s + 3.67·6-s + 1.06·7-s − 3.36·8-s − 0.523·9-s − 4.87·10-s + 4.94·11-s − 5.41·12-s − 2.48·14-s − 3.29·15-s + 0.965·16-s − 1.91·17-s + 1.22·18-s − 3.73·19-s + 7.19·20-s − 1.67·21-s − 11.5·22-s + 3.34·23-s + 5.29·24-s − 0.627·25-s + 5.54·27-s + 3.66·28-s + 4.56·29-s + 7.67·30-s + ⋯ |
| L(s) = 1 | − 1.64·2-s − 0.908·3-s + 1.72·4-s + 0.935·5-s + 1.49·6-s + 0.402·7-s − 1.18·8-s − 0.174·9-s − 1.54·10-s + 1.49·11-s − 1.56·12-s − 0.664·14-s − 0.849·15-s + 0.241·16-s − 0.463·17-s + 0.287·18-s − 0.857·19-s + 1.60·20-s − 0.365·21-s − 2.46·22-s + 0.696·23-s + 1.08·24-s − 0.125·25-s + 1.06·27-s + 0.693·28-s + 0.847·29-s + 1.40·30-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 | \( 1 + 2.33T + 2T^{2} \) |
| 3 | \( 1 + 1.57T + 3T^{2} \) |
| 5 | \( 1 - 2.09T + 5T^{2} \) |
| 7 | \( 1 - 1.06T + 7T^{2} \) |
| 11 | \( 1 - 4.94T + 11T^{2} \) |
| 17 | \( 1 + 1.91T + 17T^{2} \) |
| 19 | \( 1 + 3.73T + 19T^{2} \) |
| 23 | \( 1 - 3.34T + 23T^{2} \) |
| 29 | \( 1 - 4.56T + 29T^{2} \) |
| 37 | \( 1 - 1.32T + 37T^{2} \) |
| 41 | \( 1 + 5.01T + 41T^{2} \) |
| 43 | \( 1 + 5.31T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 - 5.11T + 53T^{2} \) |
| 59 | \( 1 - 4.18T + 59T^{2} \) |
| 61 | \( 1 - 8.82T + 61T^{2} \) |
| 67 | \( 1 + 9.43T + 67T^{2} \) |
| 71 | \( 1 + 16.7T + 71T^{2} \) |
| 73 | \( 1 - 0.634T + 73T^{2} \) |
| 79 | \( 1 + 8.49T + 79T^{2} \) |
| 83 | \( 1 - 0.137T + 83T^{2} \) |
| 89 | \( 1 + 6.40T + 89T^{2} \) |
| 97 | \( 1 - 3.31T + 97T^{2} \) |
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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
−8.137500329354197444369015849046, −6.92888696905448794566480470140, −6.61751822476374883469668496646, −6.03590064978807086088041589312, −5.12097859955540915122981167416, −4.26406923657805742928614407254, −2.86483154006717504012988908150, −1.79941368140597198951314372752, −1.23492199384907779481149444973, 0,
1.23492199384907779481149444973, 1.79941368140597198951314372752, 2.86483154006717504012988908150, 4.26406923657805742928614407254, 5.12097859955540915122981167416, 6.03590064978807086088041589312, 6.61751822476374883469668496646, 6.92888696905448794566480470140, 8.137500329354197444369015849046
