| L(s) = 1 | + 2-s − 2.98·3-s + 4-s − 2.67·5-s − 2.98·6-s + 2.38·7-s + 8-s + 5.92·9-s − 2.67·10-s − 1.89·11-s − 2.98·12-s + 6.39·13-s + 2.38·14-s + 7.98·15-s + 16-s + 0.000397·17-s + 5.92·18-s + 0.220·19-s − 2.67·20-s − 7.13·21-s − 1.89·22-s − 9.09·23-s − 2.98·24-s + 2.14·25-s + 6.39·26-s − 8.74·27-s + 2.38·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.72·3-s + 0.5·4-s − 1.19·5-s − 1.21·6-s + 0.902·7-s + 0.353·8-s + 1.97·9-s − 0.845·10-s − 0.572·11-s − 0.862·12-s + 1.77·13-s + 0.638·14-s + 2.06·15-s + 0.250·16-s + 9.64e−5·17-s + 1.39·18-s + 0.0505·19-s − 0.597·20-s − 1.55·21-s − 0.404·22-s − 1.89·23-s − 0.609·24-s + 0.428·25-s + 1.25·26-s − 1.68·27-s + 0.451·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.273726505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.273726505\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 + 2.98T + 3T^{2} \) |
| 5 | \( 1 + 2.67T + 5T^{2} \) |
| 7 | \( 1 - 2.38T + 7T^{2} \) |
| 11 | \( 1 + 1.89T + 11T^{2} \) |
| 13 | \( 1 - 6.39T + 13T^{2} \) |
| 17 | \( 1 - 0.000397T + 17T^{2} \) |
| 19 | \( 1 - 0.220T + 19T^{2} \) |
| 23 | \( 1 + 9.09T + 23T^{2} \) |
| 29 | \( 1 - 2.60T + 29T^{2} \) |
| 31 | \( 1 + 6.23T + 31T^{2} \) |
| 41 | \( 1 + 0.463T + 41T^{2} \) |
| 43 | \( 1 + 4.30T + 43T^{2} \) |
| 47 | \( 1 - 3.92T + 47T^{2} \) |
| 53 | \( 1 - 6.25T + 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 - 8.68T + 61T^{2} \) |
| 67 | \( 1 - 4.36T + 67T^{2} \) |
| 71 | \( 1 - 2.40T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 - 7.71T + 79T^{2} \) |
| 83 | \( 1 - 5.13T + 83T^{2} \) |
| 89 | \( 1 + 5.18T + 89T^{2} \) |
| 97 | \( 1 + 14.0T + 97T^{2} \) |
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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
−8.422626343706148293052327103766, −7.986934861721928997886768966278, −7.10169203784288642591662647958, −6.33248483237620228470414117366, −5.60626617345614282345899313379, −5.05255120348916036878347904664, −4.06292256926654464810674588609, −3.75058197941864849387084809802, −1.90804883922486217645951034054, −0.70131719769820914014862574499,
0.70131719769820914014862574499, 1.90804883922486217645951034054, 3.75058197941864849387084809802, 4.06292256926654464810674588609, 5.05255120348916036878347904664, 5.60626617345614282345899313379, 6.33248483237620228470414117366, 7.10169203784288642591662647958, 7.986934861721928997886768966278, 8.422626343706148293052327103766
