| L(s) = 1 | − 1.14·2-s − 3.07·3-s − 0.681·4-s − 2.50·5-s + 3.53·6-s + 3.07·8-s + 6.45·9-s + 2.87·10-s + 3.87·11-s + 2.09·12-s − 4.15·13-s + 7.71·15-s − 2.17·16-s − 4.37·17-s − 7.41·18-s + 0.683·19-s + 1.70·20-s − 4.44·22-s − 6.52·23-s − 9.46·24-s + 1.28·25-s + 4.76·26-s − 10.6·27-s − 9.24·29-s − 8.85·30-s + 31-s − 3.66·32-s + ⋯ |
| L(s) = 1 | − 0.811·2-s − 1.77·3-s − 0.340·4-s − 1.12·5-s + 1.44·6-s + 1.08·8-s + 2.15·9-s + 0.910·10-s + 1.16·11-s + 0.604·12-s − 1.15·13-s + 1.99·15-s − 0.543·16-s − 1.06·17-s − 1.74·18-s + 0.156·19-s + 0.382·20-s − 0.947·22-s − 1.36·23-s − 1.93·24-s + 0.257·25-s + 0.935·26-s − 2.04·27-s − 1.71·29-s − 1.61·30-s + 0.179·31-s − 0.647·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1250845578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1250845578\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 1.14T + 2T^{2} \) |
| 3 | \( 1 + 3.07T + 3T^{2} \) |
| 5 | \( 1 + 2.50T + 5T^{2} \) |
| 11 | \( 1 - 3.87T + 11T^{2} \) |
| 13 | \( 1 + 4.15T + 13T^{2} \) |
| 17 | \( 1 + 4.37T + 17T^{2} \) |
| 19 | \( 1 - 0.683T + 19T^{2} \) |
| 23 | \( 1 + 6.52T + 23T^{2} \) |
| 29 | \( 1 + 9.24T + 29T^{2} \) |
| 37 | \( 1 + 5.70T + 37T^{2} \) |
| 41 | \( 1 - 9.67T + 41T^{2} \) |
| 43 | \( 1 + 5.11T + 43T^{2} \) |
| 47 | \( 1 - 5.58T + 47T^{2} \) |
| 53 | \( 1 - 2.43T + 53T^{2} \) |
| 59 | \( 1 + 7.40T + 59T^{2} \) |
| 61 | \( 1 + 2.52T + 61T^{2} \) |
| 67 | \( 1 - 2.69T + 67T^{2} \) |
| 71 | \( 1 + 8.63T + 71T^{2} \) |
| 73 | \( 1 + 8.17T + 73T^{2} \) |
| 79 | \( 1 + 2.69T + 79T^{2} \) |
| 83 | \( 1 + 6.60T + 83T^{2} \) |
| 89 | \( 1 - 2.98T + 89T^{2} \) |
| 97 | \( 1 - 2.29T + 97T^{2} \) |
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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
−9.559938396716511917565775169794, −8.804208249160658523489775230468, −7.61032349410569833410259499986, −7.25513415835425136959622455446, −6.29717583102052159838961138785, −5.33066348945321721775092546026, −4.33194879970754615160020119264, −4.03642735825052010506577554518, −1.69811684157058124371411027829, −0.31412076884736577857048645136,
0.31412076884736577857048645136, 1.69811684157058124371411027829, 4.03642735825052010506577554518, 4.33194879970754615160020119264, 5.33066348945321721775092546026, 6.29717583102052159838961138785, 7.25513415835425136959622455446, 7.61032349410569833410259499986, 8.804208249160658523489775230468, 9.559938396716511917565775169794
