| L(s) = 1 | + 0.235·2-s + 3-s − 1.94·4-s + 2.04·5-s + 0.235·6-s − 0.723·7-s − 0.929·8-s + 9-s + 0.481·10-s + 11-s − 1.94·12-s − 6.69·13-s − 0.170·14-s + 2.04·15-s + 3.67·16-s − 4.95·17-s + 0.235·18-s − 3.92·19-s − 3.97·20-s − 0.723·21-s + 0.235·22-s + 5.80·23-s − 0.929·24-s − 0.821·25-s − 1.57·26-s + 27-s + 1.40·28-s + ⋯ |
| L(s) = 1 | + 0.166·2-s + 0.577·3-s − 0.972·4-s + 0.914·5-s + 0.0961·6-s − 0.273·7-s − 0.328·8-s + 0.333·9-s + 0.152·10-s + 0.301·11-s − 0.561·12-s − 1.85·13-s − 0.0455·14-s + 0.527·15-s + 0.917·16-s − 1.20·17-s + 0.0555·18-s − 0.900·19-s − 0.888·20-s − 0.157·21-s + 0.0502·22-s + 1.21·23-s − 0.189·24-s − 0.164·25-s − 0.309·26-s + 0.192·27-s + 0.265·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 0.235T + 2T^{2} \) |
| 5 | \( 1 - 2.04T + 5T^{2} \) |
| 7 | \( 1 + 0.723T + 7T^{2} \) |
| 13 | \( 1 + 6.69T + 13T^{2} \) |
| 17 | \( 1 + 4.95T + 17T^{2} \) |
| 19 | \( 1 + 3.92T + 19T^{2} \) |
| 23 | \( 1 - 5.80T + 23T^{2} \) |
| 29 | \( 1 + 2.22T + 29T^{2} \) |
| 31 | \( 1 - 1.52T + 31T^{2} \) |
| 37 | \( 1 + 9.58T + 37T^{2} \) |
| 41 | \( 1 + 6.49T + 41T^{2} \) |
| 43 | \( 1 + 9.19T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 + 0.677T + 53T^{2} \) |
| 59 | \( 1 + 0.691T + 59T^{2} \) |
| 67 | \( 1 + 1.07T + 67T^{2} \) |
| 71 | \( 1 - 9.28T + 71T^{2} \) |
| 73 | \( 1 - 8.26T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 1.81T + 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−8.954302491184345965415499165457, −8.202190550610258017050702957502, −7.09051525709183772454079829068, −6.49098110511246470648266910533, −5.27435103807456020557326768260, −4.79545988346900220571500356969, −3.81398421226729329026921628465, −2.73147831931638224937621774044, −1.83798396398534375202603919848, 0,
1.83798396398534375202603919848, 2.73147831931638224937621774044, 3.81398421226729329026921628465, 4.79545988346900220571500356969, 5.27435103807456020557326768260, 6.49098110511246470648266910533, 7.09051525709183772454079829068, 8.202190550610258017050702957502, 8.954302491184345965415499165457
