| L(s) = 1 | − 1.61·2-s + 3-s + 0.611·4-s − 2.09·5-s − 1.61·6-s + 4.41·7-s + 2.24·8-s + 9-s + 3.39·10-s + 0.611·12-s − 13-s − 7.12·14-s − 2.09·15-s − 4.84·16-s − 7.08·17-s − 1.61·18-s + 0.946·19-s − 1.28·20-s + 4.41·21-s − 4.97·23-s + 2.24·24-s − 0.592·25-s + 1.61·26-s + 27-s + 2.69·28-s + 4.98·29-s + 3.39·30-s + ⋯ |
| L(s) = 1 | − 1.14·2-s + 0.577·3-s + 0.305·4-s − 0.938·5-s − 0.659·6-s + 1.66·7-s + 0.793·8-s + 0.333·9-s + 1.07·10-s + 0.176·12-s − 0.277·13-s − 1.90·14-s − 0.542·15-s − 1.21·16-s − 1.71·17-s − 0.380·18-s + 0.217·19-s − 0.287·20-s + 0.962·21-s − 1.03·23-s + 0.457·24-s − 0.118·25-s + 0.316·26-s + 0.192·27-s + 0.509·28-s + 0.926·29-s + 0.619·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4719 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4719 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.079294467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.079294467\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 5 | \( 1 + 2.09T + 5T^{2} \) |
| 7 | \( 1 - 4.41T + 7T^{2} \) |
| 17 | \( 1 + 7.08T + 17T^{2} \) |
| 19 | \( 1 - 0.946T + 19T^{2} \) |
| 23 | \( 1 + 4.97T + 23T^{2} \) |
| 29 | \( 1 - 4.98T + 29T^{2} \) |
| 31 | \( 1 - 7.88T + 31T^{2} \) |
| 37 | \( 1 - 7.76T + 37T^{2} \) |
| 41 | \( 1 + 0.497T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 - 1.21T + 47T^{2} \) |
| 53 | \( 1 - 1.09T + 53T^{2} \) |
| 59 | \( 1 + 0.0922T + 59T^{2} \) |
| 61 | \( 1 - 9.66T + 61T^{2} \) |
| 67 | \( 1 - 8.50T + 67T^{2} \) |
| 71 | \( 1 - 16.1T + 71T^{2} \) |
| 73 | \( 1 + 3.11T + 73T^{2} \) |
| 79 | \( 1 - 2.44T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 - 7.34T + 89T^{2} \) |
| 97 | \( 1 + 7.43T + 97T^{2} \) |
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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
−8.224463950812795580132763918449, −8.046465442023617782660531800917, −7.23374133157710446295336491582, −6.51274085357827559923633738910, −5.05256172229095166011809387860, −4.46566481780139791145542848895, −3.98650401392663614603228219232, −2.49783950717939616235088505378, −1.79261663620241562993709579782, −0.67987556077205515327023495810,
0.67987556077205515327023495810, 1.79261663620241562993709579782, 2.49783950717939616235088505378, 3.98650401392663614603228219232, 4.46566481780139791145542848895, 5.05256172229095166011809387860, 6.51274085357827559923633738910, 7.23374133157710446295336491582, 8.046465442023617782660531800917, 8.224463950812795580132763918449
