L(s) = 1 | + 7-s + 4·13-s − 6·17-s − 2·19-s + 6·23-s − 2·31-s − 2·37-s + 6·41-s − 4·43-s + 49-s + 6·53-s − 12·59-s − 10·61-s − 4·67-s − 12·71-s + 4·73-s − 8·79-s + 12·83-s + 6·89-s + 4·91-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 6·119-s + ⋯ |
L(s) = 1 | + 0.377·7-s + 1.10·13-s − 1.45·17-s − 0.458·19-s + 1.25·23-s − 0.359·31-s − 0.328·37-s + 0.937·41-s − 0.609·43-s + 1/7·49-s + 0.824·53-s − 1.56·59-s − 1.28·61-s − 0.488·67-s − 1.42·71-s + 0.468·73-s − 0.900·79-s + 1.31·83-s + 0.635·89-s + 0.419·91-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.550·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−15.52133137573627, −15.13311207233201, −14.69122567877109, −13.85685043717855, −13.52493288948067, −13.01178805424512, −12.49775653166748, −11.72772260037185, −11.23891566502524, −10.72824707976540, −10.46956066257807, −9.413821565808620, −8.910208858842106, −8.664373025626433, −7.854942903905092, −7.274336451104369, −6.576984913874407, −6.148989826005222, −5.404192759346571, −4.655125136872928, −4.218524030660086, −3.408317167102460, −2.688179829594974, −1.851245820248783, −1.142383044673906, 0,
1.142383044673906, 1.851245820248783, 2.688179829594974, 3.408317167102460, 4.218524030660086, 4.655125136872928, 5.404192759346571, 6.148989826005222, 6.576984913874407, 7.274336451104369, 7.854942903905092, 8.664373025626433, 8.910208858842106, 9.413821565808620, 10.46956066257807, 10.72824707976540, 11.23891566502524, 11.72772260037185, 12.49775653166748, 13.01178805424512, 13.52493288948067, 13.85685043717855, 14.69122567877109, 15.13311207233201, 15.52133137573627