| L(s) = 1 | − 2·7-s + 2·11-s − 13-s − 2·17-s − 2·19-s + 23-s − 3·29-s + 4·31-s − 10·37-s + 2·41-s + 3·43-s − 6·47-s − 3·49-s − 11·53-s − 14·59-s − 11·61-s − 10·67-s − 2·73-s − 4·77-s − 3·79-s + 6·83-s − 12·89-s + 2·91-s − 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.603·11-s − 0.277·13-s − 0.485·17-s − 0.458·19-s + 0.208·23-s − 0.557·29-s + 0.718·31-s − 1.64·37-s + 0.312·41-s + 0.457·43-s − 0.875·47-s − 3/7·49-s − 1.51·53-s − 1.82·59-s − 1.40·61-s − 1.22·67-s − 0.234·73-s − 0.455·77-s − 0.337·79-s + 0.658·83-s − 1.27·89-s + 0.209·91-s − 1.01·97-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
| \(L(1)\) |
\(\approx\) |
\(0.2095950653\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2095950653\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.04320145629756, −12.63538500061495, −12.25945916566617, −11.80294326161193, −11.15892798699589, −10.83123957931095, −10.28137544045657, −9.757174047613197, −9.340161869679076, −8.898713761071816, −8.492800240518772, −7.723546092834211, −7.415367182726848, −6.714990865415220, −6.334985606464077, −6.038377991815922, −5.239550330290213, −4.666467190408563, −4.295840236401588, −3.515385266913981, −3.151508341695538, −2.551180539188945, −1.730262997975866, −1.316693533969688, −0.1272201344899509,
0.1272201344899509, 1.316693533969688, 1.730262997975866, 2.551180539188945, 3.151508341695538, 3.515385266913981, 4.295840236401588, 4.666467190408563, 5.239550330290213, 6.038377991815922, 6.334985606464077, 6.714990865415220, 7.415367182726848, 7.723546092834211, 8.492800240518772, 8.898713761071816, 9.340161869679076, 9.757174047613197, 10.28137544045657, 10.83123957931095, 11.15892798699589, 11.80294326161193, 12.25945916566617, 12.63538500061495, 13.04320145629756
