| L(s) = 1 | − 2·7-s − 6·11-s − 13-s − 2·17-s + 6·19-s − 6·29-s + 6·31-s + 2·37-s + 10·41-s + 8·43-s + 6·47-s − 3·49-s − 6·53-s − 6·59-s + 10·61-s + 2·67-s + 14·71-s + 14·73-s + 12·77-s + 4·79-s − 6·83-s − 6·89-s + 2·91-s + 14·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 1.80·11-s − 0.277·13-s − 0.485·17-s + 1.37·19-s − 1.11·29-s + 1.07·31-s + 0.328·37-s + 1.56·41-s + 1.21·43-s + 0.875·47-s − 3/7·49-s − 0.824·53-s − 0.781·59-s + 1.28·61-s + 0.244·67-s + 1.66·71-s + 1.63·73-s + 1.36·77-s + 0.450·79-s − 0.658·83-s − 0.635·89-s + 0.209·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.818000274\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.818000274\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.98082015979451, −12.64923071263968, −12.51275698287155, −11.56297520790263, −11.25255918245237, −10.82052982248764, −10.14161356804500, −9.940645932777916, −9.263922484771093, −9.095218634337560, −8.177381172037199, −7.716675509082495, −7.578521267594023, −6.859635170827136, −6.325794444084092, −5.699116150691589, −5.391731180566767, −4.795790041789047, −4.264391300337326, −3.496991086863636, −3.060864684362600, −2.433708517194404, −2.130195269239938, −0.9275237919582606, −0.4630880287792752,
0.4630880287792752, 0.9275237919582606, 2.130195269239938, 2.433708517194404, 3.060864684362600, 3.496991086863636, 4.264391300337326, 4.795790041789047, 5.391731180566767, 5.699116150691589, 6.325794444084092, 6.859635170827136, 7.578521267594023, 7.716675509082495, 8.177381172037199, 9.095218634337560, 9.263922484771093, 9.940645932777916, 10.14161356804500, 10.82052982248764, 11.25255918245237, 11.56297520790263, 12.51275698287155, 12.64923071263968, 12.98082015979451
