| L(s) = 1 | − 3.40·3-s − 4.31·5-s + 2.57·9-s + 17.8i·11-s + 3.25·13-s + 14.6·15-s − 15.7i·17-s + 1.55·19-s + 41.4·23-s − 6.36·25-s + 21.8·27-s + 3.74i·29-s + 0.0167i·31-s − 60.8i·33-s + 1.34i·37-s + ⋯ |
| L(s) = 1 | − 1.13·3-s − 0.863·5-s + 0.286·9-s + 1.62i·11-s + 0.250·13-s + 0.979·15-s − 0.925i·17-s + 0.0819·19-s + 1.80·23-s − 0.254·25-s + 0.809·27-s + 0.129i·29-s + 0.000540i·31-s − 1.84i·33-s + 0.0364i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.857 - 0.515i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.857 - 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3935439268\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3935439268\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 3.40T + 9T^{2} \) |
| 5 | \( 1 + 4.31T + 25T^{2} \) |
| 11 | \( 1 - 17.8iT - 121T^{2} \) |
| 13 | \( 1 - 3.25T + 169T^{2} \) |
| 17 | \( 1 + 15.7iT - 289T^{2} \) |
| 19 | \( 1 - 1.55T + 361T^{2} \) |
| 23 | \( 1 - 41.4T + 529T^{2} \) |
| 29 | \( 1 - 3.74iT - 841T^{2} \) |
| 31 | \( 1 - 0.0167iT - 961T^{2} \) |
| 37 | \( 1 - 1.34iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 70.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 13.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 35.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 45.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 68.7T + 3.48e3T^{2} \) |
| 61 | \( 1 - 96.0T + 3.72e3T^{2} \) |
| 67 | \( 1 + 13.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 75.7T + 5.04e3T^{2} \) |
| 73 | \( 1 + 53.1iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 23.3T + 6.24e3T^{2} \) |
| 83 | \( 1 + 102.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 88.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 140. iT - 9.40e3T^{2} \) |
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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
−9.589713515883117387316833913359, −8.887697651847984204525387200913, −7.69393651722337027409809767699, −7.15263194671798514667593976500, −6.44363631495951833858912547800, −5.24480521220416587247616733698, −4.81070015817124043042344991694, −3.82868534171672146473004704845, −2.56378805353425698167040675022, −1.03591749987917411858707894887,
0.17473210749664162199019782827, 1.12782855263327768640848472992, 3.00577496906905179063972604003, 3.80123620717362972797414808067, 4.88313920089538089640747782945, 5.69750786938440224491166431377, 6.33655957924532395500261356196, 7.17590404025058007087766898879, 8.331062128692303329988485183146, 8.593384622405743889873144459709
