| L(s) = 1 | − 2-s + 4-s + 1.79·5-s − 8-s − 1.79·10-s − 11-s + 3.63·13-s + 16-s − 6.11·17-s + 1.84·19-s + 1.79·20-s + 22-s + 7.37·23-s − 1.76·25-s − 3.63·26-s + 10.4·29-s + 7.97·31-s − 32-s + 6.11·34-s + 10.1·37-s − 1.84·38-s − 1.79·40-s − 8.17·41-s − 2.28·43-s − 44-s − 7.37·46-s + 0.669·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.804·5-s − 0.353·8-s − 0.568·10-s − 0.301·11-s + 1.00·13-s + 0.250·16-s − 1.48·17-s + 0.422·19-s + 0.402·20-s + 0.213·22-s + 1.53·23-s − 0.353·25-s − 0.713·26-s + 1.93·29-s + 1.43·31-s − 0.176·32-s + 1.04·34-s + 1.67·37-s − 0.298·38-s − 0.284·40-s − 1.27·41-s − 0.348·43-s − 0.150·44-s − 1.08·46-s + 0.0976·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.903187674\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.903187674\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 1.79T + 5T^{2} \) |
| 13 | \( 1 - 3.63T + 13T^{2} \) |
| 17 | \( 1 + 6.11T + 17T^{2} \) |
| 19 | \( 1 - 1.84T + 19T^{2} \) |
| 23 | \( 1 - 7.37T + 23T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 31 | \( 1 - 7.97T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 + 8.17T + 41T^{2} \) |
| 43 | \( 1 + 2.28T + 43T^{2} \) |
| 47 | \( 1 - 0.669T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 + 0.274T + 61T^{2} \) |
| 67 | \( 1 - 3.88T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 - 3.85T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 + 5.27T + 83T^{2} \) |
| 89 | \( 1 + 1.98T + 89T^{2} \) |
| 97 | \( 1 - 4.12T + 97T^{2} \) |
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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
−7.83530269542118075991057960451, −6.77025385224389762051470019992, −6.55179677144810455241336471930, −5.83023057328940260415827163622, −4.95166239804956323136239028614, −4.30128557941788803465807220847, −3.07274148683137598650688776636, −2.57349429624634832332477361136, −1.57100030283287942985611410501, −0.76948084543206307722149353370,
0.76948084543206307722149353370, 1.57100030283287942985611410501, 2.57349429624634832332477361136, 3.07274148683137598650688776636, 4.30128557941788803465807220847, 4.95166239804956323136239028614, 5.83023057328940260415827163622, 6.55179677144810455241336471930, 6.77025385224389762051470019992, 7.83530269542118075991057960451
