| L(s) = 1 | + 2.17·2-s − 1.70·3-s + 2.70·4-s + 5-s − 3.70·6-s + 1.07·7-s + 1.53·8-s − 0.0783·9-s + 2.17·10-s − 6.34·11-s − 4.63·12-s + 1.36·13-s + 2.34·14-s − 1.70·15-s − 2.07·16-s + 3.26·17-s − 0.170·18-s − 19-s + 2.70·20-s − 1.84·21-s − 13.7·22-s + 2.34·23-s − 2.63·24-s + 25-s + 2.97·26-s + 5.26·27-s + 2.92·28-s + ⋯ |
| L(s) = 1 | + 1.53·2-s − 0.986·3-s + 1.35·4-s + 0.447·5-s − 1.51·6-s + 0.407·7-s + 0.544·8-s − 0.0261·9-s + 0.686·10-s − 1.91·11-s − 1.33·12-s + 0.379·13-s + 0.625·14-s − 0.441·15-s − 0.519·16-s + 0.791·17-s − 0.0400·18-s − 0.229·19-s + 0.605·20-s − 0.402·21-s − 2.93·22-s + 0.487·23-s − 0.537·24-s + 0.200·25-s + 0.582·26-s + 1.01·27-s + 0.552·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.580626011\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.580626011\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 2.17T + 2T^{2} \) |
| 3 | \( 1 + 1.70T + 3T^{2} \) |
| 7 | \( 1 - 1.07T + 7T^{2} \) |
| 11 | \( 1 + 6.34T + 11T^{2} \) |
| 13 | \( 1 - 1.36T + 13T^{2} \) |
| 17 | \( 1 - 3.26T + 17T^{2} \) |
| 23 | \( 1 - 2.34T + 23T^{2} \) |
| 29 | \( 1 - 1.41T + 29T^{2} \) |
| 31 | \( 1 - 8.68T + 31T^{2} \) |
| 37 | \( 1 - 5.36T + 37T^{2} \) |
| 41 | \( 1 + 3.26T + 41T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 - 1.07T + 47T^{2} \) |
| 53 | \( 1 - 6.63T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 - 5.60T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 - 5.41T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 - 7.57T + 89T^{2} \) |
| 97 | \( 1 + 8.88T + 97T^{2} \) |
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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
−13.75402851145552921081864086895, −13.03977446509505532479386582771, −12.06641877440572031428002194145, −11.14160480142782964928965263681, −10.21153481149396587228071113609, −8.168712480609600236610728663562, −6.50351328755815579920507115249, −5.47236967091325229402647393917, −4.85694211108814214036381352615, −2.85801741721610885731497664984,
2.85801741721610885731497664984, 4.85694211108814214036381352615, 5.47236967091325229402647393917, 6.50351328755815579920507115249, 8.168712480609600236610728663562, 10.21153481149396587228071113609, 11.14160480142782964928965263681, 12.06641877440572031428002194145, 13.03977446509505532479386582771, 13.75402851145552921081864086895
