| L(s) = 1 | − 5-s − 0.334·7-s + 2.12·11-s − 2.25·13-s + 6.93·17-s + 4.12·19-s − 23-s + 25-s − 1.92·29-s + 0.440·31-s + 0.334·35-s + 6.04·37-s − 10.2·41-s − 5.50·43-s − 4.65·47-s − 6.88·49-s − 4.40·53-s − 2.12·55-s − 10.8·59-s + 13.7·61-s + 2.25·65-s + 6.87·67-s + 5.57·71-s + 6.25·73-s − 0.709·77-s + 16.7·79-s + 10.2·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.126·7-s + 0.639·11-s − 0.625·13-s + 1.68·17-s + 0.945·19-s − 0.208·23-s + 0.200·25-s − 0.356·29-s + 0.0791·31-s + 0.0565·35-s + 0.993·37-s − 1.60·41-s − 0.838·43-s − 0.679·47-s − 0.984·49-s − 0.604·53-s − 0.286·55-s − 1.41·59-s + 1.76·61-s + 0.279·65-s + 0.839·67-s + 0.662·71-s + 0.732·73-s − 0.0808·77-s + 1.87·79-s + 1.12·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.760595586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.760595586\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 0.334T + 7T^{2} \) |
| 11 | \( 1 - 2.12T + 11T^{2} \) |
| 13 | \( 1 + 2.25T + 13T^{2} \) |
| 17 | \( 1 - 6.93T + 17T^{2} \) |
| 19 | \( 1 - 4.12T + 19T^{2} \) |
| 29 | \( 1 + 1.92T + 29T^{2} \) |
| 31 | \( 1 - 0.440T + 31T^{2} \) |
| 37 | \( 1 - 6.04T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 + 5.50T + 43T^{2} \) |
| 47 | \( 1 + 4.65T + 47T^{2} \) |
| 53 | \( 1 + 4.40T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 - 6.87T + 67T^{2} \) |
| 71 | \( 1 - 5.57T + 71T^{2} \) |
| 73 | \( 1 - 6.25T + 73T^{2} \) |
| 79 | \( 1 - 16.7T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 - 1.33T + 89T^{2} \) |
| 97 | \( 1 - 18.4T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.133715196143899325893046810512, −7.85602317196820828078775892786, −6.97556001054172581323452503794, −6.30436981512591161886241564284, −5.32541323548661416150228161670, −4.77961163115539346300679389284, −3.58048069183240407128799693039, −3.23575419060623145544790596938, −1.88554766335286359242251120639, −0.77156465680163372332988136036,
0.77156465680163372332988136036, 1.88554766335286359242251120639, 3.23575419060623145544790596938, 3.58048069183240407128799693039, 4.77961163115539346300679389284, 5.32541323548661416150228161670, 6.30436981512591161886241564284, 6.97556001054172581323452503794, 7.85602317196820828078775892786, 8.133715196143899325893046810512
