L(s) = 1 | + 2-s − 3-s + 4-s + 1.26·5-s − 6-s + 3.87·7-s + 8-s + 9-s + 1.26·10-s − 3.70·11-s − 12-s + 6.90·13-s + 3.87·14-s − 1.26·15-s + 16-s + 17-s + 18-s + 2.11·19-s + 1.26·20-s − 3.87·21-s − 3.70·22-s + 2.51·23-s − 24-s − 3.41·25-s + 6.90·26-s − 27-s + 3.87·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.563·5-s − 0.408·6-s + 1.46·7-s + 0.353·8-s + 0.333·9-s + 0.398·10-s − 1.11·11-s − 0.288·12-s + 1.91·13-s + 1.03·14-s − 0.325·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s + 0.486·19-s + 0.281·20-s − 0.846·21-s − 0.790·22-s + 0.523·23-s − 0.204·24-s − 0.682·25-s + 1.35·26-s − 0.192·27-s + 0.732·28-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
\(L(1)\) |
\(\approx\) |
\(3.982576731\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.982576731\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.26T + 5T^{2} \) |
| 7 | \( 1 - 3.87T + 7T^{2} \) |
| 11 | \( 1 + 3.70T + 11T^{2} \) |
| 13 | \( 1 - 6.90T + 13T^{2} \) |
| 19 | \( 1 - 2.11T + 19T^{2} \) |
| 23 | \( 1 - 2.51T + 23T^{2} \) |
| 29 | \( 1 - 6.32T + 29T^{2} \) |
| 31 | \( 1 + 2.19T + 31T^{2} \) |
| 37 | \( 1 - 5.33T + 37T^{2} \) |
| 41 | \( 1 - 8.42T + 41T^{2} \) |
| 43 | \( 1 + 9.26T + 43T^{2} \) |
| 47 | \( 1 + 0.566T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 61 | \( 1 + 2.78T + 61T^{2} \) |
| 67 | \( 1 + 0.453T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 + 1.22T + 89T^{2} \) |
| 97 | \( 1 - 8.86T + 97T^{2} \) |
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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
−7.986823696148073258281092091176, −7.39525999718639230937907127538, −6.27924241129967481042698849677, −5.92252845181114004627570397822, −5.13326235112047093631269251525, −4.71775946235450469685713369234, −3.78005106336086969337446634259, −2.81694942165918954345536145394, −1.76638079464034342715751139484, −1.07940386413899735681110138458,
1.07940386413899735681110138458, 1.76638079464034342715751139484, 2.81694942165918954345536145394, 3.78005106336086969337446634259, 4.71775946235450469685713369234, 5.13326235112047093631269251525, 5.92252845181114004627570397822, 6.27924241129967481042698849677, 7.39525999718639230937907127538, 7.986823696148073258281092091176