L(s) = 1 | − 2-s − 1.67·3-s + 4-s + 1.83·5-s + 1.67·6-s + 4.34·7-s − 8-s − 0.186·9-s − 1.83·10-s − 3.12·11-s − 1.67·12-s − 1.45·13-s − 4.34·14-s − 3.08·15-s + 16-s + 7.19·17-s + 0.186·18-s − 1.42·19-s + 1.83·20-s − 7.29·21-s + 3.12·22-s − 3.84·23-s + 1.67·24-s − 1.62·25-s + 1.45·26-s + 5.34·27-s + 4.34·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.968·3-s + 0.5·4-s + 0.821·5-s + 0.684·6-s + 1.64·7-s − 0.353·8-s − 0.0622·9-s − 0.581·10-s − 0.941·11-s − 0.484·12-s − 0.404·13-s − 1.16·14-s − 0.795·15-s + 0.250·16-s + 1.74·17-s + 0.0440·18-s − 0.326·19-s + 0.410·20-s − 1.59·21-s + 0.665·22-s − 0.801·23-s + 0.342·24-s − 0.324·25-s + 0.286·26-s + 1.02·27-s + 0.821·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.181166694\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.181166694\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2003 | \( 1 - T \) |
good | 3 | \( 1 + 1.67T + 3T^{2} \) |
| 5 | \( 1 - 1.83T + 5T^{2} \) |
| 7 | \( 1 - 4.34T + 7T^{2} \) |
| 11 | \( 1 + 3.12T + 11T^{2} \) |
| 13 | \( 1 + 1.45T + 13T^{2} \) |
| 17 | \( 1 - 7.19T + 17T^{2} \) |
| 19 | \( 1 + 1.42T + 19T^{2} \) |
| 23 | \( 1 + 3.84T + 23T^{2} \) |
| 29 | \( 1 + 2.37T + 29T^{2} \) |
| 31 | \( 1 + 9.37T + 31T^{2} \) |
| 37 | \( 1 - 3.02T + 37T^{2} \) |
| 41 | \( 1 - 4.86T + 41T^{2} \) |
| 43 | \( 1 - 0.736T + 43T^{2} \) |
| 47 | \( 1 - 6.51T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 + 1.43T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 + 4.30T + 71T^{2} \) |
| 73 | \( 1 + 0.705T + 73T^{2} \) |
| 79 | \( 1 - 2.30T + 79T^{2} \) |
| 83 | \( 1 - 3.07T + 83T^{2} \) |
| 89 | \( 1 - 1.00T + 89T^{2} \) |
| 97 | \( 1 - 2.04T + 97T^{2} \) |
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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
−8.300976686098291707402523159605, −7.77845199439970739852594171309, −7.19564603631870815745480198097, −5.97138188765433628336394024911, −5.47966464967586631340723376186, −5.18635981183978640322774738496, −3.94862444275770445857706740324, −2.50555417877107929340979880332, −1.81852757278082553208862818086, −0.73730609163984389711943703582,
0.73730609163984389711943703582, 1.81852757278082553208862818086, 2.50555417877107929340979880332, 3.94862444275770445857706740324, 5.18635981183978640322774738496, 5.47966464967586631340723376186, 5.97138188765433628336394024911, 7.19564603631870815745480198097, 7.77845199439970739852594171309, 8.300976686098291707402523159605