| L(s) = 1 | − 2-s + 4-s + 3.87·5-s + 3.75·7-s − 8-s − 3.87·10-s + 2.69·11-s − 4.71·13-s − 3.75·14-s + 16-s + 5.53·17-s + 1.06·19-s + 3.87·20-s − 2.69·22-s + 3.30·23-s + 10.0·25-s + 4.71·26-s + 3.75·28-s − 0.716·29-s − 4.82·31-s − 32-s − 5.53·34-s + 14.5·35-s − 0.551·37-s − 1.06·38-s − 3.87·40-s − 2.38·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.73·5-s + 1.42·7-s − 0.353·8-s − 1.22·10-s + 0.812·11-s − 1.30·13-s − 1.00·14-s + 0.250·16-s + 1.34·17-s + 0.244·19-s + 0.867·20-s − 0.574·22-s + 0.689·23-s + 2.00·25-s + 0.925·26-s + 0.710·28-s − 0.133·29-s − 0.866·31-s − 0.176·32-s − 0.948·34-s + 2.46·35-s − 0.0906·37-s − 0.172·38-s − 0.613·40-s − 0.373·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.067777147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.067777147\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - 3.87T + 5T^{2} \) |
| 7 | \( 1 - 3.75T + 7T^{2} \) |
| 11 | \( 1 - 2.69T + 11T^{2} \) |
| 13 | \( 1 + 4.71T + 13T^{2} \) |
| 17 | \( 1 - 5.53T + 17T^{2} \) |
| 19 | \( 1 - 1.06T + 19T^{2} \) |
| 23 | \( 1 - 3.30T + 23T^{2} \) |
| 29 | \( 1 + 0.716T + 29T^{2} \) |
| 31 | \( 1 + 4.82T + 31T^{2} \) |
| 37 | \( 1 + 0.551T + 37T^{2} \) |
| 41 | \( 1 + 2.38T + 41T^{2} \) |
| 43 | \( 1 + 11.6T + 43T^{2} \) |
| 47 | \( 1 - 1.54T + 47T^{2} \) |
| 53 | \( 1 + 2.67T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 + 6.58T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 - 4.12T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 7.06T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 - 5.53T + 89T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
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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
−9.441130789889315802348913878039, −8.983653193361524013534410034858, −7.915754358667062486471196312039, −7.22987856526031233322839476527, −6.27550624991100449550167676614, −5.36057230835919548645208362095, −4.83308506231424456745404937784, −3.10280260233618169443853697140, −1.91322175677644226406406141932, −1.36326000725549227410141631487,
1.36326000725549227410141631487, 1.91322175677644226406406141932, 3.10280260233618169443853697140, 4.83308506231424456745404937784, 5.36057230835919548645208362095, 6.27550624991100449550167676614, 7.22987856526031233322839476527, 7.915754358667062486471196312039, 8.983653193361524013534410034858, 9.441130789889315802348913878039
