L(s) = 1 | + 2-s − 3-s + 4-s + 2.90·5-s − 6-s + 3.49·7-s + 8-s + 9-s + 2.90·10-s + 1.79·11-s − 12-s + 13-s + 3.49·14-s − 2.90·15-s + 16-s + 4.53·17-s + 18-s − 3.14·19-s + 2.90·20-s − 3.49·21-s + 1.79·22-s − 8.15·23-s − 24-s + 3.45·25-s + 26-s − 27-s + 3.49·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.30·5-s − 0.408·6-s + 1.32·7-s + 0.353·8-s + 0.333·9-s + 0.919·10-s + 0.542·11-s − 0.288·12-s + 0.277·13-s + 0.933·14-s − 0.750·15-s + 0.250·16-s + 1.09·17-s + 0.235·18-s − 0.721·19-s + 0.650·20-s − 0.762·21-s + 0.383·22-s − 1.70·23-s − 0.204·24-s + 0.691·25-s + 0.196·26-s − 0.192·27-s + 0.660·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.743502630\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.743502630\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
good | 5 | \( 1 - 2.90T + 5T^{2} \) |
| 7 | \( 1 - 3.49T + 7T^{2} \) |
| 11 | \( 1 - 1.79T + 11T^{2} \) |
| 17 | \( 1 - 4.53T + 17T^{2} \) |
| 19 | \( 1 + 3.14T + 19T^{2} \) |
| 23 | \( 1 + 8.15T + 23T^{2} \) |
| 29 | \( 1 - 6.89T + 29T^{2} \) |
| 31 | \( 1 + 3.14T + 31T^{2} \) |
| 37 | \( 1 + 1.41T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 + 6.04T + 43T^{2} \) |
| 47 | \( 1 - 0.558T + 47T^{2} \) |
| 53 | \( 1 - 2.33T + 53T^{2} \) |
| 59 | \( 1 - 4.35T + 59T^{2} \) |
| 61 | \( 1 - 9.67T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 - 8.34T + 73T^{2} \) |
| 79 | \( 1 - 6.31T + 79T^{2} \) |
| 83 | \( 1 + 3.92T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 + 15.0T + 97T^{2} \) |
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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
−7.87704582678330989319829005012, −6.82260871728306171338967387225, −6.26886101539864104989932874291, −5.63260783458509437609099951317, −5.23043908535643932729662481802, −4.38282668332075811464065974840, −3.76926725236822868166316542933, −2.46661515213991905944792792087, −1.81010136810805156098473998929, −1.10168956351415920156086589568,
1.10168956351415920156086589568, 1.81010136810805156098473998929, 2.46661515213991905944792792087, 3.76926725236822868166316542933, 4.38282668332075811464065974840, 5.23043908535643932729662481802, 5.63260783458509437609099951317, 6.26886101539864104989932874291, 6.82260871728306171338967387225, 7.87704582678330989319829005012