| L(s) = 1 | − 2-s + 2.54·3-s + 4-s + 5-s − 2.54·6-s + 1.88·7-s − 8-s + 3.48·9-s − 10-s − 11-s + 2.54·12-s + 4.02·13-s − 1.88·14-s + 2.54·15-s + 16-s − 1.30·17-s − 3.48·18-s + 3.46·19-s + 20-s + 4.80·21-s + 22-s + 9.00·23-s − 2.54·24-s + 25-s − 4.02·26-s + 1.22·27-s + 1.88·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.46·3-s + 0.5·4-s + 0.447·5-s − 1.03·6-s + 0.712·7-s − 0.353·8-s + 1.16·9-s − 0.316·10-s − 0.301·11-s + 0.734·12-s + 1.11·13-s − 0.503·14-s + 0.657·15-s + 0.250·16-s − 0.317·17-s − 0.820·18-s + 0.795·19-s + 0.223·20-s + 1.04·21-s + 0.213·22-s + 1.87·23-s − 0.519·24-s + 0.200·25-s − 0.788·26-s + 0.235·27-s + 0.356·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.242752088\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.242752088\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 - 2.54T + 3T^{2} \) |
| 7 | \( 1 - 1.88T + 7T^{2} \) |
| 13 | \( 1 - 4.02T + 13T^{2} \) |
| 17 | \( 1 + 1.30T + 17T^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 23 | \( 1 - 9.00T + 23T^{2} \) |
| 29 | \( 1 - 2.56T + 29T^{2} \) |
| 31 | \( 1 + 0.578T + 31T^{2} \) |
| 37 | \( 1 + 3.03T + 37T^{2} \) |
| 41 | \( 1 - 7.32T + 41T^{2} \) |
| 47 | \( 1 + 5.84T + 47T^{2} \) |
| 53 | \( 1 + 1.34T + 53T^{2} \) |
| 59 | \( 1 + 7.46T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 + 3.85T + 67T^{2} \) |
| 71 | \( 1 - 3.78T + 71T^{2} \) |
| 73 | \( 1 - 7.39T + 73T^{2} \) |
| 79 | \( 1 - 9.14T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 + 15.2T + 89T^{2} \) |
| 97 | \( 1 - 8.82T + 97T^{2} \) |
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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
−8.321874648970160068070603403343, −7.85391059619292560291611842895, −7.14554441498093664382588002822, −6.34395513238517859439416482345, −5.35499916639986790451172716167, −4.49949721520994429049231743196, −3.33678463760419593860687093283, −2.86829719642425185448687729073, −1.85046617272342965768742068651, −1.14262104522903966403114556769,
1.14262104522903966403114556769, 1.85046617272342965768742068651, 2.86829719642425185448687729073, 3.33678463760419593860687093283, 4.49949721520994429049231743196, 5.35499916639986790451172716167, 6.34395513238517859439416482345, 7.14554441498093664382588002822, 7.85391059619292560291611842895, 8.321874648970160068070603403343
