L(s) = 1 | − 0.182·2-s + 3-s − 1.96·4-s − 1.12·5-s − 0.182·6-s + 7-s + 0.722·8-s + 9-s + 0.205·10-s − 4.01·11-s − 1.96·12-s − 0.182·14-s − 1.12·15-s + 3.80·16-s + 3.02·17-s − 0.182·18-s − 5.30·19-s + 2.22·20-s + 21-s + 0.731·22-s − 2.17·23-s + 0.722·24-s − 3.72·25-s + 27-s − 1.96·28-s + 3.53·29-s + 0.205·30-s + ⋯ |
L(s) = 1 | − 0.128·2-s + 0.577·3-s − 0.983·4-s − 0.504·5-s − 0.0743·6-s + 0.377·7-s + 0.255·8-s + 0.333·9-s + 0.0650·10-s − 1.21·11-s − 0.567·12-s − 0.0486·14-s − 0.291·15-s + 0.950·16-s + 0.733·17-s − 0.0429·18-s − 1.21·19-s + 0.496·20-s + 0.218·21-s + 0.155·22-s − 0.453·23-s + 0.147·24-s − 0.745·25-s + 0.192·27-s − 0.371·28-s + 0.656·29-s + 0.0375·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.245029098\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.245029098\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 0.182T + 2T^{2} \) |
| 5 | \( 1 + 1.12T + 5T^{2} \) |
| 11 | \( 1 + 4.01T + 11T^{2} \) |
| 17 | \( 1 - 3.02T + 17T^{2} \) |
| 19 | \( 1 + 5.30T + 19T^{2} \) |
| 23 | \( 1 + 2.17T + 23T^{2} \) |
| 29 | \( 1 - 3.53T + 29T^{2} \) |
| 31 | \( 1 - 4.85T + 31T^{2} \) |
| 37 | \( 1 - 6.38T + 37T^{2} \) |
| 41 | \( 1 + 4.76T + 41T^{2} \) |
| 43 | \( 1 + 2.34T + 43T^{2} \) |
| 47 | \( 1 - 9.81T + 47T^{2} \) |
| 53 | \( 1 - 2.98T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 - 4.70T + 61T^{2} \) |
| 67 | \( 1 - 1.77T + 67T^{2} \) |
| 71 | \( 1 + 3.24T + 71T^{2} \) |
| 73 | \( 1 - 7.80T + 73T^{2} \) |
| 79 | \( 1 - 17.4T + 79T^{2} \) |
| 83 | \( 1 - 0.705T + 83T^{2} \) |
| 89 | \( 1 + 13.0T + 89T^{2} \) |
| 97 | \( 1 + 1.76T + 97T^{2} \) |
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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
−8.386056651592590311556711999438, −7.996983751543627621787322157492, −7.51545036666649224436562759767, −6.29305563250012048746523172056, −5.39205295136327130661474492800, −4.59879916279255561561513991105, −4.01679329303119315340416911457, −3.07753592999501655292748896427, −2.06571673968903413777282541730, −0.64640304403840329637477127475,
0.64640304403840329637477127475, 2.06571673968903413777282541730, 3.07753592999501655292748896427, 4.01679329303119315340416911457, 4.59879916279255561561513991105, 5.39205295136327130661474492800, 6.29305563250012048746523172056, 7.51545036666649224436562759767, 7.996983751543627621787322157492, 8.386056651592590311556711999438