| L(s) = 1 | − 2-s + 3-s + 4-s + 2.24·5-s − 6-s + 7-s − 8-s + 9-s − 2.24·10-s + 1.30·11-s + 12-s − 14-s + 2.24·15-s + 16-s + 4.93·17-s − 18-s + 1.08·19-s + 2.24·20-s + 21-s − 1.30·22-s − 4.85·23-s − 24-s + 0.0489·25-s + 27-s + 28-s + 1.04·29-s − 2.24·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.00·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.710·10-s + 0.394·11-s + 0.288·12-s − 0.267·14-s + 0.580·15-s + 0.250·16-s + 1.19·17-s − 0.235·18-s + 0.249·19-s + 0.502·20-s + 0.218·21-s − 0.278·22-s − 1.01·23-s − 0.204·24-s + 0.00978·25-s + 0.192·27-s + 0.188·28-s + 0.194·29-s − 0.410·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.673097589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.673097589\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 2.24T + 5T^{2} \) |
| 11 | \( 1 - 1.30T + 11T^{2} \) |
| 17 | \( 1 - 4.93T + 17T^{2} \) |
| 19 | \( 1 - 1.08T + 19T^{2} \) |
| 23 | \( 1 + 4.85T + 23T^{2} \) |
| 29 | \( 1 - 1.04T + 29T^{2} \) |
| 31 | \( 1 - 5.29T + 31T^{2} \) |
| 37 | \( 1 - 0.149T + 37T^{2} \) |
| 41 | \( 1 - 4.24T + 41T^{2} \) |
| 43 | \( 1 - 3.07T + 43T^{2} \) |
| 47 | \( 1 + 3.31T + 47T^{2} \) |
| 53 | \( 1 + 1.29T + 53T^{2} \) |
| 59 | \( 1 - 0.862T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 - 3.04T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 - 0.567T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 - 1.35T + 83T^{2} \) |
| 89 | \( 1 - 7.66T + 89T^{2} \) |
| 97 | \( 1 - 9.27T + 97T^{2} \) |
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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
−8.017703891263035276732176505971, −7.46647826738916155795960592340, −6.56595933013212344082838108443, −5.96052750724829171700237646764, −5.28113676643355385446190640593, −4.27052756952124668685060278622, −3.36756623665607615454563569592, −2.48754870539949231738289710042, −1.76557171647060461215918920766, −0.953133143676382665755213183411,
0.953133143676382665755213183411, 1.76557171647060461215918920766, 2.48754870539949231738289710042, 3.36756623665607615454563569592, 4.27052756952124668685060278622, 5.28113676643355385446190640593, 5.96052750724829171700237646764, 6.56595933013212344082838108443, 7.46647826738916155795960592340, 8.017703891263035276732176505971
