L(s) = 1 | − 2-s + 0.271·3-s + 4-s − 5-s − 0.271·6-s + 1.75·7-s − 8-s − 2.92·9-s + 10-s + 0.0291·11-s + 0.271·12-s + 5.25·13-s − 1.75·14-s − 0.271·15-s + 16-s − 3.06·17-s + 2.92·18-s − 6.99·19-s − 20-s + 0.477·21-s − 0.0291·22-s + 3.94·23-s − 0.271·24-s + 25-s − 5.25·26-s − 1.60·27-s + 1.75·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.156·3-s + 0.5·4-s − 0.447·5-s − 0.110·6-s + 0.665·7-s − 0.353·8-s − 0.975·9-s + 0.316·10-s + 0.00878·11-s + 0.0783·12-s + 1.45·13-s − 0.470·14-s − 0.0701·15-s + 0.250·16-s − 0.744·17-s + 0.689·18-s − 1.60·19-s − 0.223·20-s + 0.104·21-s − 0.00620·22-s + 0.823·23-s − 0.0554·24-s + 0.200·25-s − 1.03·26-s − 0.309·27-s + 0.332·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
good | 3 | \( 1 - 0.271T + 3T^{2} \) |
| 7 | \( 1 - 1.75T + 7T^{2} \) |
| 11 | \( 1 - 0.0291T + 11T^{2} \) |
| 13 | \( 1 - 5.25T + 13T^{2} \) |
| 17 | \( 1 + 3.06T + 17T^{2} \) |
| 19 | \( 1 + 6.99T + 19T^{2} \) |
| 23 | \( 1 - 3.94T + 23T^{2} \) |
| 29 | \( 1 - 8.01T + 29T^{2} \) |
| 31 | \( 1 + 7.61T + 31T^{2} \) |
| 37 | \( 1 + 3.48T + 37T^{2} \) |
| 41 | \( 1 - 1.23T + 41T^{2} \) |
| 43 | \( 1 - 3.83T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 - 1.28T + 59T^{2} \) |
| 61 | \( 1 + 5.25T + 61T^{2} \) |
| 67 | \( 1 + 2.24T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 + 2.43T + 79T^{2} \) |
| 83 | \( 1 - 8.86T + 83T^{2} \) |
| 89 | \( 1 - 12.0T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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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
−8.332189134220904406137302653508, −7.56329516745542786610030002023, −6.60623707583315976647944706807, −6.11248685454691414789994068140, −5.09509773860663945414231139701, −4.18810977554811713739587018087, −3.30501885960915770936615965949, −2.35117619110972442694153945615, −1.34151703321977202334308445314, 0,
1.34151703321977202334308445314, 2.35117619110972442694153945615, 3.30501885960915770936615965949, 4.18810977554811713739587018087, 5.09509773860663945414231139701, 6.11248685454691414789994068140, 6.60623707583315976647944706807, 7.56329516745542786610030002023, 8.332189134220904406137302653508