L(s) = 1 | − 2-s − 3-s + 4-s + 0.539·5-s + 6-s + 0.211·7-s − 8-s + 9-s − 0.539·10-s − 0.438·11-s − 12-s − 2.12·13-s − 0.211·14-s − 0.539·15-s + 16-s − 17-s − 18-s + 1.46·19-s + 0.539·20-s − 0.211·21-s + 0.438·22-s + 5.56·23-s + 24-s − 4.70·25-s + 2.12·26-s − 27-s + 0.211·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.241·5-s + 0.408·6-s + 0.0799·7-s − 0.353·8-s + 0.333·9-s − 0.170·10-s − 0.132·11-s − 0.288·12-s − 0.588·13-s − 0.0565·14-s − 0.139·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s + 0.336·19-s + 0.120·20-s − 0.0461·21-s + 0.0935·22-s + 1.16·23-s + 0.204·24-s − 0.941·25-s + 0.416·26-s − 0.192·27-s + 0.0399·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
good | 5 | \( 1 - 0.539T + 5T^{2} \) |
| 7 | \( 1 - 0.211T + 7T^{2} \) |
| 11 | \( 1 + 0.438T + 11T^{2} \) |
| 13 | \( 1 + 2.12T + 13T^{2} \) |
| 19 | \( 1 - 1.46T + 19T^{2} \) |
| 23 | \( 1 - 5.56T + 23T^{2} \) |
| 29 | \( 1 + 4.54T + 29T^{2} \) |
| 31 | \( 1 - 4.41T + 31T^{2} \) |
| 37 | \( 1 - 6.05T + 37T^{2} \) |
| 41 | \( 1 - 3.13T + 41T^{2} \) |
| 43 | \( 1 + 12.5T + 43T^{2} \) |
| 47 | \( 1 + 3.68T + 47T^{2} \) |
| 53 | \( 1 - 3.44T + 53T^{2} \) |
| 61 | \( 1 - 7.40T + 61T^{2} \) |
| 67 | \( 1 + 5.96T + 67T^{2} \) |
| 71 | \( 1 - 16.5T + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 + 1.40T + 79T^{2} \) |
| 83 | \( 1 + 5.92T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−7.84310004664509987381008319811, −6.91671105640227764990184911516, −6.54583547530481688493428111020, −5.55235987337957764036533668684, −5.06108168478020966941346372074, −4.09203744569519830786427191814, −3.04106535901092829190483917574, −2.14306842617079346435990071801, −1.17197778325730170127525979617, 0,
1.17197778325730170127525979617, 2.14306842617079346435990071801, 3.04106535901092829190483917574, 4.09203744569519830786427191814, 5.06108168478020966941346372074, 5.55235987337957764036533668684, 6.54583547530481688493428111020, 6.91671105640227764990184911516, 7.84310004664509987381008319811