L(s) = 1 | + 2-s + 4-s − 2·7-s + 8-s − 3.46·11-s + 1.46·13-s − 2·14-s + 16-s + 3.46·17-s + 19-s − 3.46·22-s − 6.92·23-s + 1.46·26-s − 2·28-s + 3.46·29-s + 5.46·31-s + 32-s + 3.46·34-s − 5.46·37-s + 38-s + 1.46·43-s − 3.46·44-s − 6.92·46-s − 6.92·47-s − 3·49-s + 1.46·52-s + 0.928·53-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.755·7-s + 0.353·8-s − 1.04·11-s + 0.406·13-s − 0.534·14-s + 0.250·16-s + 0.840·17-s + 0.229·19-s − 0.738·22-s − 1.44·23-s + 0.287·26-s − 0.377·28-s + 0.643·29-s + 0.981·31-s + 0.176·32-s + 0.594·34-s − 0.898·37-s + 0.162·38-s + 0.223·43-s − 0.522·44-s − 1.02·46-s − 1.01·47-s − 0.428·49-s + 0.203·52-s + 0.127·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 - 1.46T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 23 | \( 1 + 6.92T + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 - 5.46T + 31T^{2} \) |
| 37 | \( 1 + 5.46T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 1.46T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 - 0.928T + 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 1.07T + 67T^{2} \) |
| 71 | \( 1 + 6.92T + 71T^{2} \) |
| 73 | \( 1 + 8.92T + 73T^{2} \) |
| 79 | \( 1 - 5.46T + 79T^{2} \) |
| 83 | \( 1 + 9.46T + 83T^{2} \) |
| 89 | \( 1 - 6.92T + 89T^{2} \) |
| 97 | \( 1 - 2.39T + 97T^{2} \) |
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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
−7.42827890411515433264187551021, −6.52491785595371188323623664064, −6.06051260545352622197328154707, −5.36524180923671289793688489375, −4.67721492299394773183167644550, −3.79355622689864536053840083075, −3.14254415834998264270871275610, −2.48286079645468487884422033925, −1.37796362315432232441698483175, 0,
1.37796362315432232441698483175, 2.48286079645468487884422033925, 3.14254415834998264270871275610, 3.79355622689864536053840083075, 4.67721492299394773183167644550, 5.36524180923671289793688489375, 6.06051260545352622197328154707, 6.52491785595371188323623664064, 7.42827890411515433264187551021