| L(s) = 1 | + 2-s + 2.30·3-s + 4-s + 3.15·5-s + 2.30·6-s − 7-s + 8-s + 2.31·9-s + 3.15·10-s − 2.86·11-s + 2.30·12-s + 6.74·13-s − 14-s + 7.26·15-s + 16-s − 3.61·17-s + 2.31·18-s + 3.15·20-s − 2.30·21-s − 2.86·22-s + 9.19·23-s + 2.30·24-s + 4.93·25-s + 6.74·26-s − 1.58·27-s − 28-s − 3.62·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.33·3-s + 0.5·4-s + 1.40·5-s + 0.941·6-s − 0.377·7-s + 0.353·8-s + 0.771·9-s + 0.996·10-s − 0.863·11-s + 0.665·12-s + 1.87·13-s − 0.267·14-s + 1.87·15-s + 0.250·16-s − 0.877·17-s + 0.545·18-s + 0.704·20-s − 0.503·21-s − 0.610·22-s + 1.91·23-s + 0.470·24-s + 0.986·25-s + 1.32·26-s − 0.304·27-s − 0.188·28-s − 0.673·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.603134358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.603134358\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 2.30T + 3T^{2} \) |
| 5 | \( 1 - 3.15T + 5T^{2} \) |
| 11 | \( 1 + 2.86T + 11T^{2} \) |
| 13 | \( 1 - 6.74T + 13T^{2} \) |
| 17 | \( 1 + 3.61T + 17T^{2} \) |
| 23 | \( 1 - 9.19T + 23T^{2} \) |
| 29 | \( 1 + 3.62T + 29T^{2} \) |
| 31 | \( 1 - 0.789T + 31T^{2} \) |
| 37 | \( 1 + 8.66T + 37T^{2} \) |
| 41 | \( 1 - 1.81T + 41T^{2} \) |
| 43 | \( 1 - 5.87T + 43T^{2} \) |
| 47 | \( 1 - 5.19T + 47T^{2} \) |
| 53 | \( 1 - 0.209T + 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 - 1.83T + 61T^{2} \) |
| 67 | \( 1 + 15.0T + 67T^{2} \) |
| 71 | \( 1 + 6.92T + 71T^{2} \) |
| 73 | \( 1 + 6.67T + 73T^{2} \) |
| 79 | \( 1 + 1.08T + 79T^{2} \) |
| 83 | \( 1 + 8.02T + 83T^{2} \) |
| 89 | \( 1 + 5.21T + 89T^{2} \) |
| 97 | \( 1 - 15.7T + 97T^{2} \) |
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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
−8.514366572565379911060564019925, −7.37597991961273798526271948386, −6.79389759196704213442188662768, −5.87247570302663585031157033847, −5.51387915035665797712233201913, −4.41490661429140413345965212211, −3.50209617169940595015736074326, −2.87413120421570406222440793668, −2.21003860126543491646618266564, −1.33851264462515067686884104439,
1.33851264462515067686884104439, 2.21003860126543491646618266564, 2.87413120421570406222440793668, 3.50209617169940595015736074326, 4.41490661429140413345965212211, 5.51387915035665797712233201913, 5.87247570302663585031157033847, 6.79389759196704213442188662768, 7.37597991961273798526271948386, 8.514366572565379911060564019925
