L(s) = 1 | − 2-s + 1.42·3-s + 4-s − 0.00740·5-s − 1.42·6-s − 4.01·7-s − 8-s − 0.969·9-s + 0.00740·10-s + 2.42·11-s + 1.42·12-s + 2.69·13-s + 4.01·14-s − 0.0105·15-s + 16-s − 5.36·17-s + 0.969·18-s + 2.51·19-s − 0.00740·20-s − 5.71·21-s − 2.42·22-s + 0.461·23-s − 1.42·24-s − 4.99·25-s − 2.69·26-s − 5.65·27-s − 4.01·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.822·3-s + 0.5·4-s − 0.00331·5-s − 0.581·6-s − 1.51·7-s − 0.353·8-s − 0.323·9-s + 0.00234·10-s + 0.731·11-s + 0.411·12-s + 0.746·13-s + 1.07·14-s − 0.00272·15-s + 0.250·16-s − 1.30·17-s + 0.228·18-s + 0.577·19-s − 0.00165·20-s − 1.24·21-s − 0.517·22-s + 0.0962·23-s − 0.290·24-s − 0.999·25-s − 0.527·26-s − 1.08·27-s − 0.757·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.315181452\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.315181452\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2017 | \( 1 - T \) |
good | 3 | \( 1 - 1.42T + 3T^{2} \) |
| 5 | \( 1 + 0.00740T + 5T^{2} \) |
| 7 | \( 1 + 4.01T + 7T^{2} \) |
| 11 | \( 1 - 2.42T + 11T^{2} \) |
| 13 | \( 1 - 2.69T + 13T^{2} \) |
| 17 | \( 1 + 5.36T + 17T^{2} \) |
| 19 | \( 1 - 2.51T + 19T^{2} \) |
| 23 | \( 1 - 0.461T + 23T^{2} \) |
| 29 | \( 1 - 4.13T + 29T^{2} \) |
| 31 | \( 1 - 9.77T + 31T^{2} \) |
| 37 | \( 1 + 6.09T + 37T^{2} \) |
| 41 | \( 1 - 8.11T + 41T^{2} \) |
| 43 | \( 1 - 7.49T + 43T^{2} \) |
| 47 | \( 1 + 8.02T + 47T^{2} \) |
| 53 | \( 1 - 5.12T + 53T^{2} \) |
| 59 | \( 1 + 2.90T + 59T^{2} \) |
| 61 | \( 1 + 7.29T + 61T^{2} \) |
| 67 | \( 1 + 3.26T + 67T^{2} \) |
| 71 | \( 1 - 4.63T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + 2.16T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 - 16.6T + 97T^{2} \) |
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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
−8.623885712180083981331490786376, −7.899073391286340865304701361843, −7.00040682632349943004222983429, −6.33178119200159001869293972498, −5.91014975279706455101660515222, −4.41958968790056957448491172477, −3.52186643882297952301426005912, −2.94000098245698534111819955857, −2.06510979834045479497444985145, −0.67806761757348902693806908088,
0.67806761757348902693806908088, 2.06510979834045479497444985145, 2.94000098245698534111819955857, 3.52186643882297952301426005912, 4.41958968790056957448491172477, 5.91014975279706455101660515222, 6.33178119200159001869293972498, 7.00040682632349943004222983429, 7.899073391286340865304701361843, 8.623885712180083981331490786376