| L(s) = 1 | − 1.83·2-s + 1.36·4-s + 2.25·5-s − 0.744·7-s + 1.16·8-s − 4.14·10-s − 6.30·11-s − 1.64·13-s + 1.36·14-s − 4.86·16-s − 4.16·17-s + 1.04·19-s + 3.08·20-s + 11.5·22-s + 4.57·23-s + 0.100·25-s + 3.01·26-s − 1.01·28-s − 3.53·31-s + 6.60·32-s + 7.64·34-s − 1.68·35-s − 1.07·37-s − 1.90·38-s + 2.62·40-s + 1.20·41-s − 12.8·43-s + ⋯ |
| L(s) = 1 | − 1.29·2-s + 0.683·4-s + 1.01·5-s − 0.281·7-s + 0.411·8-s − 1.31·10-s − 1.90·11-s − 0.456·13-s + 0.364·14-s − 1.21·16-s − 1.01·17-s + 0.238·19-s + 0.689·20-s + 2.46·22-s + 0.953·23-s + 0.0201·25-s + 0.592·26-s − 0.192·28-s − 0.634·31-s + 1.16·32-s + 1.31·34-s − 0.284·35-s − 0.176·37-s − 0.309·38-s + 0.415·40-s + 0.187·41-s − 1.95·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5691872699\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5691872699\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + 1.83T + 2T^{2} \) |
| 5 | \( 1 - 2.25T + 5T^{2} \) |
| 7 | \( 1 + 0.744T + 7T^{2} \) |
| 11 | \( 1 + 6.30T + 11T^{2} \) |
| 13 | \( 1 + 1.64T + 13T^{2} \) |
| 17 | \( 1 + 4.16T + 17T^{2} \) |
| 19 | \( 1 - 1.04T + 19T^{2} \) |
| 23 | \( 1 - 4.57T + 23T^{2} \) |
| 31 | \( 1 + 3.53T + 31T^{2} \) |
| 37 | \( 1 + 1.07T + 37T^{2} \) |
| 41 | \( 1 - 1.20T + 41T^{2} \) |
| 43 | \( 1 + 12.8T + 43T^{2} \) |
| 47 | \( 1 - 7.88T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 6.44T + 61T^{2} \) |
| 67 | \( 1 + 6.87T + 67T^{2} \) |
| 71 | \( 1 - 3.27T + 71T^{2} \) |
| 73 | \( 1 - 7.09T + 73T^{2} \) |
| 79 | \( 1 - 6.38T + 79T^{2} \) |
| 83 | \( 1 - 5.15T + 83T^{2} \) |
| 89 | \( 1 + 9.31T + 89T^{2} \) |
| 97 | \( 1 + 15.1T + 97T^{2} \) |
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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
−7.979826318912860520544191456228, −7.31666833259966518142164801982, −6.78112065584042566795983837303, −5.83264945669465112251807418350, −5.14798609332655718258099791678, −4.56217328703907305928760827145, −3.15851709053142453697335680255, −2.35446360535370328444529632957, −1.76814497460022526539607501279, −0.44373090913206291325624697759,
0.44373090913206291325624697759, 1.76814497460022526539607501279, 2.35446360535370328444529632957, 3.15851709053142453697335680255, 4.56217328703907305928760827145, 5.14798609332655718258099791678, 5.83264945669465112251807418350, 6.78112065584042566795983837303, 7.31666833259966518142164801982, 7.979826318912860520544191456228
