| L(s) = 1 | − 1.05·2-s + 3-s − 0.876·4-s − 3.52·5-s − 1.05·6-s − 1.13·7-s + 3.04·8-s + 9-s + 3.73·10-s − 2.54·11-s − 0.876·12-s + 13-s + 1.20·14-s − 3.52·15-s − 1.47·16-s − 7.38·17-s − 1.05·18-s + 8.00·19-s + 3.09·20-s − 1.13·21-s + 2.69·22-s + 2.92·23-s + 3.04·24-s + 7.44·25-s − 1.05·26-s + 27-s + 0.994·28-s + ⋯ |
| L(s) = 1 | − 0.749·2-s + 0.577·3-s − 0.438·4-s − 1.57·5-s − 0.432·6-s − 0.428·7-s + 1.07·8-s + 0.333·9-s + 1.18·10-s − 0.767·11-s − 0.253·12-s + 0.277·13-s + 0.321·14-s − 0.910·15-s − 0.369·16-s − 1.79·17-s − 0.249·18-s + 1.83·19-s + 0.691·20-s − 0.247·21-s + 0.574·22-s + 0.610·23-s + 0.622·24-s + 1.48·25-s − 0.207·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4313215784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4313215784\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 1.05T + 2T^{2} \) |
| 5 | \( 1 + 3.52T + 5T^{2} \) |
| 7 | \( 1 + 1.13T + 7T^{2} \) |
| 11 | \( 1 + 2.54T + 11T^{2} \) |
| 17 | \( 1 + 7.38T + 17T^{2} \) |
| 19 | \( 1 - 8.00T + 19T^{2} \) |
| 23 | \( 1 - 2.92T + 23T^{2} \) |
| 29 | \( 1 + 4.25T + 29T^{2} \) |
| 31 | \( 1 + 4.59T + 31T^{2} \) |
| 37 | \( 1 + 3.74T + 37T^{2} \) |
| 41 | \( 1 + 4.39T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 + 3.97T + 47T^{2} \) |
| 53 | \( 1 + 8.46T + 53T^{2} \) |
| 59 | \( 1 + 6.10T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 - 3.97T + 67T^{2} \) |
| 71 | \( 1 - 2.94T + 71T^{2} \) |
| 73 | \( 1 + 4.87T + 73T^{2} \) |
| 79 | \( 1 + 9.14T + 79T^{2} \) |
| 83 | \( 1 - 16.2T + 83T^{2} \) |
| 89 | \( 1 - 2.27T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−8.503822221802808866472970949318, −7.79166045685387723076533761236, −7.35121894925240368508638480711, −6.66138691564952882210355779176, −5.13821774513814174720555632127, −4.66550208741491169149784817282, −3.60511344491947741571957867047, −3.24053708973763767711801170386, −1.79209607107300160225838482217, −0.40555574426818322246168361773,
0.40555574426818322246168361773, 1.79209607107300160225838482217, 3.24053708973763767711801170386, 3.60511344491947741571957867047, 4.66550208741491169149784817282, 5.13821774513814174720555632127, 6.66138691564952882210355779176, 7.35121894925240368508638480711, 7.79166045685387723076533761236, 8.503822221802808866472970949318
