| L(s) = 1 | − 7-s + 3·11-s − 13-s + 3·17-s + 4·19-s + 6·23-s + 3·29-s + 31-s − 2·37-s − 10·43-s + 3·47-s − 6·49-s + 3·53-s − 15·59-s − 13·61-s − 13·67-s + 10·73-s − 3·77-s + 4·79-s − 15·83-s − 6·89-s + 91-s + 4·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.904·11-s − 0.277·13-s + 0.727·17-s + 0.917·19-s + 1.25·23-s + 0.557·29-s + 0.179·31-s − 0.328·37-s − 1.52·43-s + 0.437·47-s − 6/7·49-s + 0.412·53-s − 1.95·59-s − 1.66·61-s − 1.58·67-s + 1.17·73-s − 0.341·77-s + 0.450·79-s − 1.64·83-s − 0.635·89-s + 0.104·91-s + 0.406·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−14.82416287075866, −14.33611570936798, −13.79030466393304, −13.46374565347974, −12.68530576000081, −12.25444187689223, −11.86593884504940, −11.26680885924296, −10.75161669284646, −9.995225406736726, −9.745593318097353, −9.047505431322494, −8.717585269156694, −7.917254318026098, −7.355962629850493, −6.922114649053482, −6.232377133413806, −5.823236823233250, −4.843731081848754, −4.745302859980027, −3.632469355240183, −3.255267646642553, −2.673481456207800, −1.548460694754005, −1.116896409377772, 0,
1.116896409377772, 1.548460694754005, 2.673481456207800, 3.255267646642553, 3.632469355240183, 4.745302859980027, 4.843731081848754, 5.823236823233250, 6.232377133413806, 6.922114649053482, 7.355962629850493, 7.917254318026098, 8.717585269156694, 9.047505431322494, 9.745593318097353, 9.995225406736726, 10.75161669284646, 11.26680885924296, 11.86593884504940, 12.25444187689223, 12.68530576000081, 13.46374565347974, 13.79030466393304, 14.33611570936798, 14.82416287075866
