L(s) = 1 | − 5-s − 5·11-s − 2·13-s + 6·17-s − 2·19-s + 6·23-s − 4·25-s + 3·29-s − 5·31-s − 2·37-s + 8·41-s − 4·43-s + 4·47-s − 9·53-s + 5·55-s − 3·59-s + 12·61-s + 2·65-s + 2·67-s + 8·71-s + 14·73-s + 79-s + 17·83-s − 6·85-s + 18·89-s + 2·95-s − 3·97-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 1.50·11-s − 0.554·13-s + 1.45·17-s − 0.458·19-s + 1.25·23-s − 4/5·25-s + 0.557·29-s − 0.898·31-s − 0.328·37-s + 1.24·41-s − 0.609·43-s + 0.583·47-s − 1.23·53-s + 0.674·55-s − 0.390·59-s + 1.53·61-s + 0.248·65-s + 0.244·67-s + 0.949·71-s + 1.63·73-s + 0.112·79-s + 1.86·83-s − 0.650·85-s + 1.90·89-s + 0.205·95-s − 0.304·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.313364347\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.313364347\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−8.369918761466630880298894604371, −7.77771197292795900244599361068, −7.34304996635694380096313509714, −6.33942017831400936799505022289, −5.32671410671974924764523850134, −4.99021165093168935165425234307, −3.81956195187003903134398581317, −3.02836498850616989097954644520, −2.13761465123081612427304168009, −0.65546615564291070885369388619,
0.65546615564291070885369388619, 2.13761465123081612427304168009, 3.02836498850616989097954644520, 3.81956195187003903134398581317, 4.99021165093168935165425234307, 5.32671410671974924764523850134, 6.33942017831400936799505022289, 7.34304996635694380096313509714, 7.77771197292795900244599361068, 8.369918761466630880298894604371