L(s) = 1 | + 2-s + 1.40·3-s + 4-s + 5-s + 1.40·6-s − 1.57·7-s + 8-s − 1.03·9-s + 10-s + 2.98·11-s + 1.40·12-s + 13-s − 1.57·14-s + 1.40·15-s + 16-s + 0.689·17-s − 1.03·18-s + 6.64·19-s + 20-s − 2.20·21-s + 2.98·22-s − 4.04·23-s + 1.40·24-s + 25-s + 26-s − 5.65·27-s − 1.57·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.809·3-s + 0.5·4-s + 0.447·5-s + 0.572·6-s − 0.595·7-s + 0.353·8-s − 0.345·9-s + 0.316·10-s + 0.901·11-s + 0.404·12-s + 0.277·13-s − 0.420·14-s + 0.361·15-s + 0.250·16-s + 0.167·17-s − 0.244·18-s + 1.52·19-s + 0.223·20-s − 0.481·21-s + 0.637·22-s − 0.843·23-s + 0.286·24-s + 0.200·25-s + 0.196·26-s − 1.08·27-s − 0.297·28-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
\(L(1)\) |
\(\approx\) |
\(4.470516994\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.470516994\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.40T + 3T^{2} \) |
| 7 | \( 1 + 1.57T + 7T^{2} \) |
| 11 | \( 1 - 2.98T + 11T^{2} \) |
| 17 | \( 1 - 0.689T + 17T^{2} \) |
| 19 | \( 1 - 6.64T + 19T^{2} \) |
| 23 | \( 1 + 4.04T + 23T^{2} \) |
| 29 | \( 1 - 5.90T + 29T^{2} \) |
| 37 | \( 1 - 4.02T + 37T^{2} \) |
| 41 | \( 1 + 0.671T + 41T^{2} \) |
| 43 | \( 1 - 5.19T + 43T^{2} \) |
| 47 | \( 1 - 9.40T + 47T^{2} \) |
| 53 | \( 1 - 4.54T + 53T^{2} \) |
| 59 | \( 1 - 5.23T + 59T^{2} \) |
| 61 | \( 1 - 3.98T + 61T^{2} \) |
| 67 | \( 1 - 1.17T + 67T^{2} \) |
| 71 | \( 1 + 7.19T + 71T^{2} \) |
| 73 | \( 1 + 8.84T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 + 4.78T + 83T^{2} \) |
| 89 | \( 1 + 7.93T + 89T^{2} \) |
| 97 | \( 1 - 18.0T + 97T^{2} \) |
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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
−8.520491487548759499619053565768, −7.60950336234137554698251160510, −6.96142985949054516773368607665, −6.00151768058704527169873677564, −5.69608009968563536481017384206, −4.52144240868796200538853134504, −3.67994705418320720790064156343, −3.06083114550173837848787425653, −2.29204944535458433918314013731, −1.11255364174549810131606520676,
1.11255364174549810131606520676, 2.29204944535458433918314013731, 3.06083114550173837848787425653, 3.67994705418320720790064156343, 4.52144240868796200538853134504, 5.69608009968563536481017384206, 6.00151768058704527169873677564, 6.96142985949054516773368607665, 7.60950336234137554698251160510, 8.520491487548759499619053565768