L(s) = 1 | + 2-s − 3.42·3-s + 4-s + 5-s − 3.42·6-s + 1.86·7-s + 8-s + 8.75·9-s + 10-s − 3.60·11-s − 3.42·12-s + 2.83·13-s + 1.86·14-s − 3.42·15-s + 16-s + 5.98·17-s + 8.75·18-s + 20-s − 6.40·21-s − 3.60·22-s + 5.08·23-s − 3.42·24-s + 25-s + 2.83·26-s − 19.7·27-s + 1.86·28-s − 1.17·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.97·3-s + 0.5·4-s + 0.447·5-s − 1.39·6-s + 0.705·7-s + 0.353·8-s + 2.91·9-s + 0.316·10-s − 1.08·11-s − 0.989·12-s + 0.786·13-s + 0.498·14-s − 0.885·15-s + 0.250·16-s + 1.45·17-s + 2.06·18-s + 0.223·20-s − 1.39·21-s − 0.768·22-s + 1.06·23-s − 0.699·24-s + 0.200·25-s + 0.556·26-s − 3.79·27-s + 0.352·28-s − 0.218·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.992429661\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.992429661\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + 3.42T + 3T^{2} \) |
| 7 | \( 1 - 1.86T + 7T^{2} \) |
| 11 | \( 1 + 3.60T + 11T^{2} \) |
| 13 | \( 1 - 2.83T + 13T^{2} \) |
| 17 | \( 1 - 5.98T + 17T^{2} \) |
| 23 | \( 1 - 5.08T + 23T^{2} \) |
| 29 | \( 1 + 1.17T + 29T^{2} \) |
| 31 | \( 1 - 5.19T + 31T^{2} \) |
| 37 | \( 1 + 3.35T + 37T^{2} \) |
| 41 | \( 1 - 3.72T + 41T^{2} \) |
| 43 | \( 1 - 0.671T + 43T^{2} \) |
| 47 | \( 1 + 6.32T + 47T^{2} \) |
| 53 | \( 1 + 2.56T + 53T^{2} \) |
| 59 | \( 1 - 4.76T + 59T^{2} \) |
| 61 | \( 1 - 8.23T + 61T^{2} \) |
| 67 | \( 1 + 7.96T + 67T^{2} \) |
| 71 | \( 1 - 1.92T + 71T^{2} \) |
| 73 | \( 1 + 5.01T + 73T^{2} \) |
| 79 | \( 1 - 0.601T + 79T^{2} \) |
| 83 | \( 1 + 1.90T + 83T^{2} \) |
| 89 | \( 1 + 0.985T + 89T^{2} \) |
| 97 | \( 1 - 0.187T + 97T^{2} \) |
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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
−8.287983574308722445708260115257, −7.51490531669972374941912498578, −6.80866363461683693996922252927, −6.01829786601293143833182063903, −5.45990785188307262443499870601, −5.04407224742762676664611364156, −4.31525136134895935727000306092, −3.16282398614624596116231376999, −1.72558907068914447466801760113, −0.884756369936596766114495567642,
0.884756369936596766114495567642, 1.72558907068914447466801760113, 3.16282398614624596116231376999, 4.31525136134895935727000306092, 5.04407224742762676664611364156, 5.45990785188307262443499870601, 6.01829786601293143833182063903, 6.80866363461683693996922252927, 7.51490531669972374941912498578, 8.287983574308722445708260115257