L(s) = 1 | − 2-s + 1.61·3-s + 4-s + 2.42·5-s − 1.61·6-s − 0.973·7-s − 8-s − 0.407·9-s − 2.42·10-s + 2.60·11-s + 1.61·12-s − 6.69·13-s + 0.973·14-s + 3.90·15-s + 16-s − 1.11·17-s + 0.407·18-s − 0.775·19-s + 2.42·20-s − 1.56·21-s − 2.60·22-s − 4.30·23-s − 1.61·24-s + 0.874·25-s + 6.69·26-s − 5.48·27-s − 0.973·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.929·3-s + 0.5·4-s + 1.08·5-s − 0.657·6-s − 0.367·7-s − 0.353·8-s − 0.135·9-s − 0.766·10-s + 0.784·11-s + 0.464·12-s − 1.85·13-s + 0.260·14-s + 1.00·15-s + 0.250·16-s − 0.269·17-s + 0.0959·18-s − 0.177·19-s + 0.541·20-s − 0.341·21-s − 0.554·22-s − 0.897·23-s − 0.328·24-s + 0.174·25-s + 1.31·26-s − 1.05·27-s − 0.183·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2011 | \( 1 + T \) |
good | 3 | \( 1 - 1.61T + 3T^{2} \) |
| 5 | \( 1 - 2.42T + 5T^{2} \) |
| 7 | \( 1 + 0.973T + 7T^{2} \) |
| 11 | \( 1 - 2.60T + 11T^{2} \) |
| 13 | \( 1 + 6.69T + 13T^{2} \) |
| 17 | \( 1 + 1.11T + 17T^{2} \) |
| 19 | \( 1 + 0.775T + 19T^{2} \) |
| 23 | \( 1 + 4.30T + 23T^{2} \) |
| 29 | \( 1 + 4.39T + 29T^{2} \) |
| 31 | \( 1 + 1.21T + 31T^{2} \) |
| 37 | \( 1 - 6.23T + 37T^{2} \) |
| 41 | \( 1 - 1.15T + 41T^{2} \) |
| 43 | \( 1 - 5.76T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 + 0.764T + 53T^{2} \) |
| 59 | \( 1 + 0.714T + 59T^{2} \) |
| 61 | \( 1 + 8.19T + 61T^{2} \) |
| 67 | \( 1 + 2.52T + 67T^{2} \) |
| 71 | \( 1 + 8.44T + 71T^{2} \) |
| 73 | \( 1 - 16.7T + 73T^{2} \) |
| 79 | \( 1 + 6.47T + 79T^{2} \) |
| 83 | \( 1 - 9.33T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 + 6.56T + 97T^{2} \) |
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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
−8.040967890430777832939860218425, −7.61716499097149768069154775606, −6.65026302786250069117343574480, −6.08016652762513573403191785902, −5.20344918964566479477108636923, −4.11155919095180346733896533307, −3.04871688323334585149962451200, −2.31788177227682915202264343124, −1.72866188206254325555408906118, 0,
1.72866188206254325555408906118, 2.31788177227682915202264343124, 3.04871688323334585149962451200, 4.11155919095180346733896533307, 5.20344918964566479477108636923, 6.08016652762513573403191785902, 6.65026302786250069117343574480, 7.61716499097149768069154775606, 8.040967890430777832939860218425