| L(s) = 1 | − 2-s − 2.96·3-s + 4-s + 0.539·5-s + 2.96·6-s − 0.220·7-s − 8-s + 5.79·9-s − 0.539·10-s + 5.11·11-s − 2.96·12-s − 1.83·13-s + 0.220·14-s − 1.59·15-s + 16-s + 4.75·17-s − 5.79·18-s − 4.79·19-s + 0.539·20-s + 0.652·21-s − 5.11·22-s + 2.67·23-s + 2.96·24-s − 4.70·25-s + 1.83·26-s − 8.28·27-s − 0.220·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.71·3-s + 0.5·4-s + 0.241·5-s + 1.21·6-s − 0.0831·7-s − 0.353·8-s + 1.93·9-s − 0.170·10-s + 1.54·11-s − 0.856·12-s − 0.510·13-s + 0.0588·14-s − 0.412·15-s + 0.250·16-s + 1.15·17-s − 1.36·18-s − 1.10·19-s + 0.120·20-s + 0.142·21-s − 1.08·22-s + 0.557·23-s + 0.605·24-s − 0.941·25-s + 0.360·26-s − 1.59·27-s − 0.0415·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2003 | \( 1 + T \) |
| good | 3 | \( 1 + 2.96T + 3T^{2} \) |
| 5 | \( 1 - 0.539T + 5T^{2} \) |
| 7 | \( 1 + 0.220T + 7T^{2} \) |
| 11 | \( 1 - 5.11T + 11T^{2} \) |
| 13 | \( 1 + 1.83T + 13T^{2} \) |
| 17 | \( 1 - 4.75T + 17T^{2} \) |
| 19 | \( 1 + 4.79T + 19T^{2} \) |
| 23 | \( 1 - 2.67T + 23T^{2} \) |
| 29 | \( 1 + 5.41T + 29T^{2} \) |
| 31 | \( 1 + 6.62T + 31T^{2} \) |
| 37 | \( 1 + 1.59T + 37T^{2} \) |
| 41 | \( 1 + 4.76T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 - 2.31T + 47T^{2} \) |
| 53 | \( 1 - 2.96T + 53T^{2} \) |
| 59 | \( 1 + 4.09T + 59T^{2} \) |
| 61 | \( 1 + 2.30T + 61T^{2} \) |
| 67 | \( 1 - 4.94T + 67T^{2} \) |
| 71 | \( 1 - 5.87T + 71T^{2} \) |
| 73 | \( 1 + 4.62T + 73T^{2} \) |
| 79 | \( 1 + 9.45T + 79T^{2} \) |
| 83 | \( 1 - 7.62T + 83T^{2} \) |
| 89 | \( 1 - 9.82T + 89T^{2} \) |
| 97 | \( 1 + 3.10T + 97T^{2} \) |
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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
−7.937010514660902968891820748615, −7.14858258414877665024332789739, −6.62304793472048623505990051530, −5.89290388154380578054752822767, −5.42607997923189079613914394840, −4.36663505790314702427280639088, −3.57597125348805448605710371146, −1.96949717104001575526532156619, −1.14995734287139036213914705274, 0,
1.14995734287139036213914705274, 1.96949717104001575526532156619, 3.57597125348805448605710371146, 4.36663505790314702427280639088, 5.42607997923189079613914394840, 5.89290388154380578054752822767, 6.62304793472048623505990051530, 7.14858258414877665024332789739, 7.937010514660902968891820748615
