L(s) = 1 | − 2-s + 2.99·3-s + 4-s − 3.99·5-s − 2.99·6-s − 3.11·7-s − 8-s + 5.95·9-s + 3.99·10-s − 0.497·11-s + 2.99·12-s + 0.868·13-s + 3.11·14-s − 11.9·15-s + 16-s + 5.07·17-s − 5.95·18-s − 2.60·19-s − 3.99·20-s − 9.31·21-s + 0.497·22-s + 8.77·23-s − 2.99·24-s + 10.9·25-s − 0.868·26-s + 8.84·27-s − 3.11·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.72·3-s + 0.5·4-s − 1.78·5-s − 1.22·6-s − 1.17·7-s − 0.353·8-s + 1.98·9-s + 1.26·10-s − 0.149·11-s + 0.863·12-s + 0.240·13-s + 0.831·14-s − 3.08·15-s + 0.250·16-s + 1.23·17-s − 1.40·18-s − 0.597·19-s − 0.892·20-s − 2.03·21-s + 0.106·22-s + 1.82·23-s − 0.610·24-s + 2.18·25-s − 0.170·26-s + 1.70·27-s − 0.588·28-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 | \( 1 - 2.99T + 3T^{2} \) |
| 5 | \( 1 + 3.99T + 5T^{2} \) |
| 7 | \( 1 + 3.11T + 7T^{2} \) |
| 11 | \( 1 + 0.497T + 11T^{2} \) |
| 13 | \( 1 - 0.868T + 13T^{2} \) |
| 17 | \( 1 - 5.07T + 17T^{2} \) |
| 19 | \( 1 + 2.60T + 19T^{2} \) |
| 23 | \( 1 - 8.77T + 23T^{2} \) |
| 29 | \( 1 + 7.43T + 29T^{2} \) |
| 31 | \( 1 + 7.56T + 31T^{2} \) |
| 37 | \( 1 - 8.87T + 37T^{2} \) |
| 41 | \( 1 - 6.44T + 41T^{2} \) |
| 43 | \( 1 + 6.92T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 + 7.49T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 - 0.465T + 67T^{2} \) |
| 71 | \( 1 - 6.79T + 71T^{2} \) |
| 73 | \( 1 + 5.92T + 73T^{2} \) |
| 79 | \( 1 - 0.651T + 79T^{2} \) |
| 83 | \( 1 + 3.28T + 83T^{2} \) |
| 89 | \( 1 - 1.48T + 89T^{2} \) |
| 97 | \( 1 + 7.62T + 97T^{2} \) |
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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
−8.075726311953561439555989052465, −7.52018224222788690093134600459, −7.19832203185666620626470019796, −6.21201344282824803741691555249, −4.76460799381436622387267498080, −3.72790664125561473149632087816, −3.32370796359246665410788652112, −2.81275736940098117005256792951, −1.38653636497791612816791349590, 0,
1.38653636497791612816791349590, 2.81275736940098117005256792951, 3.32370796359246665410788652112, 3.72790664125561473149632087816, 4.76460799381436622387267498080, 6.21201344282824803741691555249, 7.19832203185666620626470019796, 7.52018224222788690093134600459, 8.075726311953561439555989052465