| L(s) = 1 | − 2.50·2-s − 0.598·3-s + 4.28·4-s + 1.50·6-s + 3.52·7-s − 5.73·8-s − 2.64·9-s − 3.05·11-s − 2.56·12-s − 5.62·13-s − 8.84·14-s + 5.80·16-s + 6.62·18-s − 4.57·19-s − 2.10·21-s + 7.67·22-s + 0.455·23-s + 3.43·24-s + 14.1·26-s + 3.37·27-s + 15.1·28-s − 9.46·29-s − 4.24·31-s − 3.08·32-s + 1.83·33-s − 11.3·36-s − 9.85·37-s + ⋯ |
| L(s) = 1 | − 1.77·2-s − 0.345·3-s + 2.14·4-s + 0.612·6-s + 1.33·7-s − 2.02·8-s − 0.880·9-s − 0.922·11-s − 0.740·12-s − 1.56·13-s − 2.36·14-s + 1.45·16-s + 1.56·18-s − 1.04·19-s − 0.460·21-s + 1.63·22-s + 0.0950·23-s + 0.700·24-s + 2.76·26-s + 0.649·27-s + 2.85·28-s − 1.75·29-s − 0.761·31-s − 0.545·32-s + 0.318·33-s − 1.88·36-s − 1.61·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1785569924\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1785569924\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 2.50T + 2T^{2} \) |
| 3 | \( 1 + 0.598T + 3T^{2} \) |
| 7 | \( 1 - 3.52T + 7T^{2} \) |
| 11 | \( 1 + 3.05T + 11T^{2} \) |
| 13 | \( 1 + 5.62T + 13T^{2} \) |
| 19 | \( 1 + 4.57T + 19T^{2} \) |
| 23 | \( 1 - 0.455T + 23T^{2} \) |
| 29 | \( 1 + 9.46T + 29T^{2} \) |
| 31 | \( 1 + 4.24T + 31T^{2} \) |
| 37 | \( 1 + 9.85T + 37T^{2} \) |
| 41 | \( 1 - 0.505T + 41T^{2} \) |
| 43 | \( 1 + 9.70T + 43T^{2} \) |
| 47 | \( 1 - 4.72T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 - 3.04T + 67T^{2} \) |
| 71 | \( 1 + 4.20T + 71T^{2} \) |
| 73 | \( 1 + 8.48T + 73T^{2} \) |
| 79 | \( 1 + 3.66T + 79T^{2} \) |
| 83 | \( 1 - 2.54T + 83T^{2} \) |
| 89 | \( 1 - 4.72T + 89T^{2} \) |
| 97 | \( 1 - 3.05T + 97T^{2} \) |
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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
−7.909502441011540681425552971186, −7.51276378198306468971108546178, −6.95718983287103717657097946941, −5.88652648098976793443583308106, −5.26187910472460682744525288539, −4.58679678316274458137891903437, −3.13696403665558313576796729257, −2.15973579869590210862237403102, −1.78996295334555753138915200501, −0.26891196719410154404863182842,
0.26891196719410154404863182842, 1.78996295334555753138915200501, 2.15973579869590210862237403102, 3.13696403665558313576796729257, 4.58679678316274458137891903437, 5.26187910472460682744525288539, 5.88652648098976793443583308106, 6.95718983287103717657097946941, 7.51276378198306468971108546178, 7.909502441011540681425552971186
