| L(s) = 1 | + 1.19·2-s − 0.583·4-s − 2.59·5-s − 2.14·7-s − 3.07·8-s − 3.08·10-s − 0.255·11-s + 5.10·13-s − 2.55·14-s − 2.49·16-s − 5.25·17-s + 1.51·20-s − 0.303·22-s − 8.78·23-s + 1.73·25-s + 6.08·26-s + 1.25·28-s + 0.322·29-s + 2.21·31-s + 3.18·32-s − 6.25·34-s + 5.56·35-s − 4.63·37-s + 7.97·40-s + 12.0·41-s + 9.03·43-s + 0.148·44-s + ⋯ |
| L(s) = 1 | + 0.841·2-s − 0.291·4-s − 1.16·5-s − 0.810·7-s − 1.08·8-s − 0.976·10-s − 0.0769·11-s + 1.41·13-s − 0.682·14-s − 0.623·16-s − 1.27·17-s + 0.338·20-s − 0.0647·22-s − 1.83·23-s + 0.346·25-s + 1.19·26-s + 0.236·28-s + 0.0598·29-s + 0.397·31-s + 0.562·32-s − 1.07·34-s + 0.940·35-s − 0.762·37-s + 1.26·40-s + 1.87·41-s + 1.37·43-s + 0.0224·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.112545130\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.112545130\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 1.19T + 2T^{2} \) |
| 5 | \( 1 + 2.59T + 5T^{2} \) |
| 7 | \( 1 + 2.14T + 7T^{2} \) |
| 11 | \( 1 + 0.255T + 11T^{2} \) |
| 13 | \( 1 - 5.10T + 13T^{2} \) |
| 17 | \( 1 + 5.25T + 17T^{2} \) |
| 23 | \( 1 + 8.78T + 23T^{2} \) |
| 29 | \( 1 - 0.322T + 29T^{2} \) |
| 31 | \( 1 - 2.21T + 31T^{2} \) |
| 37 | \( 1 + 4.63T + 37T^{2} \) |
| 41 | \( 1 - 12.0T + 41T^{2} \) |
| 43 | \( 1 - 9.03T + 43T^{2} \) |
| 47 | \( 1 + 3.43T + 47T^{2} \) |
| 53 | \( 1 - 4.73T + 53T^{2} \) |
| 59 | \( 1 - 1.47T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 - 5.25T + 79T^{2} \) |
| 83 | \( 1 + 15.2T + 83T^{2} \) |
| 89 | \( 1 - 2.31T + 89T^{2} \) |
| 97 | \( 1 - 7.03T + 97T^{2} \) |
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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
−8.534798770847130536411306282538, −8.054885386642371768276384797175, −6.97998903903641506056760705026, −6.21556156080945892700833037988, −5.70019980293214434426018344356, −4.46279189636660418858202070925, −3.97377374356156591909944348459, −3.48875882415105397253381949804, −2.38193423614255943672245735663, −0.53537241382600306898279680055,
0.53537241382600306898279680055, 2.38193423614255943672245735663, 3.48875882415105397253381949804, 3.97377374356156591909944348459, 4.46279189636660418858202070925, 5.70019980293214434426018344356, 6.21556156080945892700833037988, 6.97998903903641506056760705026, 8.054885386642371768276384797175, 8.534798770847130536411306282538
