L(s) = 1 | − 2-s + 0.00478·3-s + 4-s − 1.04·5-s − 0.00478·6-s − 7-s − 8-s − 2.99·9-s + 1.04·10-s − 4.80·11-s + 0.00478·12-s − 3.58·13-s + 14-s − 0.00497·15-s + 16-s − 6.75·17-s + 2.99·18-s − 1.04·20-s − 0.00478·21-s + 4.80·22-s + 0.904·23-s − 0.00478·24-s − 3.91·25-s + 3.58·26-s − 0.0287·27-s − 28-s − 3.14·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.00276·3-s + 0.5·4-s − 0.465·5-s − 0.00195·6-s − 0.377·7-s − 0.353·8-s − 0.999·9-s + 0.328·10-s − 1.44·11-s + 0.00138·12-s − 0.994·13-s + 0.267·14-s − 0.00128·15-s + 0.250·16-s − 1.63·17-s + 0.707·18-s − 0.232·20-s − 0.00104·21-s + 1.02·22-s + 0.188·23-s − 0.000976·24-s − 0.783·25-s + 0.703·26-s − 0.00552·27-s − 0.188·28-s − 0.584·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.05855206362\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.05855206362\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 - 0.00478T + 3T^{2} \) |
| 5 | \( 1 + 1.04T + 5T^{2} \) |
| 11 | \( 1 + 4.80T + 11T^{2} \) |
| 13 | \( 1 + 3.58T + 13T^{2} \) |
| 17 | \( 1 + 6.75T + 17T^{2} \) |
| 23 | \( 1 - 0.904T + 23T^{2} \) |
| 29 | \( 1 + 3.14T + 29T^{2} \) |
| 31 | \( 1 + 2.02T + 31T^{2} \) |
| 37 | \( 1 + 9.28T + 37T^{2} \) |
| 41 | \( 1 + 8.12T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 - 8.48T + 47T^{2} \) |
| 53 | \( 1 + 8.76T + 53T^{2} \) |
| 59 | \( 1 - 1.50T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 + 8.96T + 67T^{2} \) |
| 71 | \( 1 + 8.59T + 71T^{2} \) |
| 73 | \( 1 - 13.5T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 0.247T + 83T^{2} \) |
| 89 | \( 1 + 3.50T + 89T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
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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
−8.227913181777495054524184307879, −7.57899487932291126990397557198, −7.05620486258240764011408032640, −6.12774828559190400444138511809, −5.39347094706267338226837616377, −4.64093956525895803894817530468, −3.51451153377914709264976836960, −2.64999263848230237145121001354, −2.06569571516420468216385612774, −0.13304057559465360881475282462,
0.13304057559465360881475282462, 2.06569571516420468216385612774, 2.64999263848230237145121001354, 3.51451153377914709264976836960, 4.64093956525895803894817530468, 5.39347094706267338226837616377, 6.12774828559190400444138511809, 7.05620486258240764011408032640, 7.57899487932291126990397557198, 8.227913181777495054524184307879