L(s) = 1 | − 7-s − 4·11-s − 6·13-s + 6·17-s + 4·23-s − 6·29-s + 2·37-s − 2·41-s − 4·43-s − 4·47-s + 49-s + 6·53-s − 12·59-s + 10·61-s − 4·67-s + 8·71-s + 14·73-s + 4·77-s − 8·79-s − 12·83-s + 6·89-s + 6·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
L(s) = 1 | − 0.377·7-s − 1.20·11-s − 1.66·13-s + 1.45·17-s + 0.834·23-s − 1.11·29-s + 0.328·37-s − 0.312·41-s − 0.609·43-s − 0.583·47-s + 1/7·49-s + 0.824·53-s − 1.56·59-s + 1.28·61-s − 0.488·67-s + 0.949·71-s + 1.63·73-s + 0.455·77-s − 0.900·79-s − 1.31·83-s + 0.635·89-s + 0.628·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 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 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.98871777026993, −13.45960520873383, −12.79607255272711, −12.72367749359711, −12.09135662458391, −11.61846174706405, −11.03046654723440, −10.42799775015072, −10.02886065119166, −9.646390913500967, −9.191528611892488, −8.441415356659871, −7.817193348099529, −7.582629235693824, −7.036955725825645, −6.485981641803324, −5.633803671885383, −5.294483800413331, −4.939831715523025, −4.177530888411601, −3.392328538664539, −2.955612703990040, −2.398491820105141, −1.705837281456300, −0.7236360151609874, 0,
0.7236360151609874, 1.705837281456300, 2.398491820105141, 2.955612703990040, 3.392328538664539, 4.177530888411601, 4.939831715523025, 5.294483800413331, 5.633803671885383, 6.485981641803324, 7.036955725825645, 7.582629235693824, 7.817193348099529, 8.441415356659871, 9.191528611892488, 9.646390913500967, 10.02886065119166, 10.42799775015072, 11.03046654723440, 11.61846174706405, 12.09135662458391, 12.72367749359711, 12.79607255272711, 13.45960520873383, 13.98871777026993