L(s) = 1 | − 2-s + 3.23·3-s + 4-s − 5-s − 3.23·6-s − 3.85·7-s − 8-s + 7.47·9-s + 10-s − 5.23·11-s + 3.23·12-s + 13-s + 3.85·14-s − 3.23·15-s + 16-s + 2.38·17-s − 7.47·18-s + 4.09·19-s − 20-s − 12.4·21-s + 5.23·22-s − 5.85·23-s − 3.23·24-s + 25-s − 26-s + 14.4·27-s − 3.85·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.86·3-s + 0.5·4-s − 0.447·5-s − 1.32·6-s − 1.45·7-s − 0.353·8-s + 2.49·9-s + 0.316·10-s − 1.57·11-s + 0.934·12-s + 0.277·13-s + 1.03·14-s − 0.835·15-s + 0.250·16-s + 0.577·17-s − 1.76·18-s + 0.938·19-s − 0.223·20-s − 2.72·21-s + 1.11·22-s − 1.22·23-s − 0.660·24-s + 0.200·25-s − 0.196·26-s + 2.78·27-s − 0.728·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.023992382\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.023992382\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
good | 3 | \( 1 - 3.23T + 3T^{2} \) |
| 7 | \( 1 + 3.85T + 7T^{2} \) |
| 11 | \( 1 + 5.23T + 11T^{2} \) |
| 17 | \( 1 - 2.38T + 17T^{2} \) |
| 19 | \( 1 - 4.09T + 19T^{2} \) |
| 23 | \( 1 + 5.85T + 23T^{2} \) |
| 29 | \( 1 - 9.61T + 29T^{2} \) |
| 37 | \( 1 - 1.09T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 - 0.763T + 43T^{2} \) |
| 47 | \( 1 - 0.0901T + 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 - 0.145T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 - 8.47T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 1.23T + 73T^{2} \) |
| 79 | \( 1 - 7.52T + 79T^{2} \) |
| 83 | \( 1 - 4.85T + 83T^{2} \) |
| 89 | \( 1 - 4.14T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−8.179341830867882837611224862542, −8.041697403633572916401856560805, −7.33558156980303238954473259120, −6.60784931159788051487961453598, −5.59461382075425144420489285261, −4.30025659602895525412134912434, −3.42297614653404103395128370945, −2.89244802660586351583907479712, −2.30136046449261034835897949647, −0.807348533499459926017870989737,
0.807348533499459926017870989737, 2.30136046449261034835897949647, 2.89244802660586351583907479712, 3.42297614653404103395128370945, 4.30025659602895525412134912434, 5.59461382075425144420489285261, 6.60784931159788051487961453598, 7.33558156980303238954473259120, 8.041697403633572916401856560805, 8.179341830867882837611224862542