L(s) = 1 | − 2-s − 3-s + 4-s − 2.83·5-s + 6-s − 4.03·7-s − 8-s + 9-s + 2.83·10-s + 1.63·11-s − 12-s + 2.83·13-s + 4.03·14-s + 2.83·15-s + 16-s + 6.03·17-s − 18-s + 19-s − 2.83·20-s + 4.03·21-s − 1.63·22-s + 5.46·23-s + 24-s + 3.03·25-s − 2.83·26-s − 27-s − 4.03·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.26·5-s + 0.408·6-s − 1.52·7-s − 0.353·8-s + 0.333·9-s + 0.896·10-s + 0.493·11-s − 0.288·12-s + 0.786·13-s + 1.07·14-s + 0.731·15-s + 0.250·16-s + 1.46·17-s − 0.235·18-s + 0.229·19-s − 0.633·20-s + 0.880·21-s − 0.348·22-s + 1.14·23-s + 0.204·24-s + 0.606·25-s − 0.555·26-s − 0.192·27-s − 0.762·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5358906521\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5358906521\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 + T \) |
good | 5 | \( 1 + 2.83T + 5T^{2} \) |
| 7 | \( 1 + 4.03T + 7T^{2} \) |
| 11 | \( 1 - 1.63T + 11T^{2} \) |
| 13 | \( 1 - 2.83T + 13T^{2} \) |
| 17 | \( 1 - 6.03T + 17T^{2} \) |
| 23 | \( 1 - 5.46T + 23T^{2} \) |
| 29 | \( 1 + 8.62T + 29T^{2} \) |
| 31 | \( 1 - 1.27T + 31T^{2} \) |
| 37 | \( 1 - 3.63T + 37T^{2} \) |
| 41 | \( 1 + 0.867T + 41T^{2} \) |
| 43 | \( 1 + 5.46T + 43T^{2} \) |
| 47 | \( 1 + 7.39T + 47T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 - 0.768T + 67T^{2} \) |
| 71 | \( 1 - 1.60T + 71T^{2} \) |
| 73 | \( 1 + 0.761T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 + 7.93T + 83T^{2} \) |
| 89 | \( 1 + 8.70T + 89T^{2} \) |
| 97 | \( 1 + 9.13T + 97T^{2} \) |
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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
−8.031143165374386301761376939637, −7.29796671364614912397572497093, −6.85334179343506122103541039265, −6.07317935469097713842482463199, −5.44994343887715437629214696224, −4.24181121667867412414242092575, −3.43829666412722359614515297414, −3.11580301221858800608677010678, −1.41643573621987703465073744020, −0.47307858427833738323096286378,
0.47307858427833738323096286378, 1.41643573621987703465073744020, 3.11580301221858800608677010678, 3.43829666412722359614515297414, 4.24181121667867412414242092575, 5.44994343887715437629214696224, 6.07317935469097713842482463199, 6.85334179343506122103541039265, 7.29796671364614912397572497093, 8.031143165374386301761376939637