| L(s) = 1 | − 2-s + 3.10·3-s + 4-s − 5-s − 3.10·6-s − 0.764·7-s − 8-s + 6.66·9-s + 10-s + 4.68·11-s + 3.10·12-s + 13-s + 0.764·14-s − 3.10·15-s + 16-s + 5.41·17-s − 6.66·18-s + 1.27·19-s − 20-s − 2.37·21-s − 4.68·22-s + 4.12·23-s − 3.10·24-s + 25-s − 26-s + 11.3·27-s − 0.764·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.79·3-s + 0.5·4-s − 0.447·5-s − 1.26·6-s − 0.288·7-s − 0.353·8-s + 2.22·9-s + 0.316·10-s + 1.41·11-s + 0.897·12-s + 0.277·13-s + 0.204·14-s − 0.802·15-s + 0.250·16-s + 1.31·17-s − 1.57·18-s + 0.293·19-s − 0.223·20-s − 0.518·21-s − 0.998·22-s + 0.859·23-s − 0.634·24-s + 0.200·25-s − 0.196·26-s + 2.19·27-s − 0.144·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.963614098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.963614098\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 3 | \( 1 - 3.10T + 3T^{2} \) |
| 7 | \( 1 + 0.764T + 7T^{2} \) |
| 11 | \( 1 - 4.68T + 11T^{2} \) |
| 17 | \( 1 - 5.41T + 17T^{2} \) |
| 19 | \( 1 - 1.27T + 19T^{2} \) |
| 23 | \( 1 - 4.12T + 23T^{2} \) |
| 29 | \( 1 + 5.56T + 29T^{2} \) |
| 37 | \( 1 + 3.38T + 37T^{2} \) |
| 41 | \( 1 + 7.29T + 41T^{2} \) |
| 43 | \( 1 - 8.33T + 43T^{2} \) |
| 47 | \( 1 + 3.98T + 47T^{2} \) |
| 53 | \( 1 + 7.62T + 53T^{2} \) |
| 59 | \( 1 - 6.82T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 - 16.3T + 67T^{2} \) |
| 71 | \( 1 - 2.74T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 + 14.0T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−8.552653094950425178690551419454, −7.897986836338615439436642273814, −7.17575443656808239226556036385, −6.74959313734584335518479098010, −5.54472946016896985106806600380, −4.24989629109061746307007266263, −3.47186723111053240349505726341, −3.12607249233368904552485027104, −1.88282606264114408955276462970, −1.10576739820353253970147769783,
1.10576739820353253970147769783, 1.88282606264114408955276462970, 3.12607249233368904552485027104, 3.47186723111053240349505726341, 4.24989629109061746307007266263, 5.54472946016896985106806600380, 6.74959313734584335518479098010, 7.17575443656808239226556036385, 7.897986836338615439436642273814, 8.552653094950425178690551419454
