| L(s) = 1 | + 2-s + 4-s − 1.03·7-s + 8-s + 3.86·11-s + 1.03·13-s − 1.03·14-s + 16-s − 7.46·17-s − 19-s + 3.86·22-s + 1.46·23-s + 1.03·26-s − 1.03·28-s − 9.52·29-s + 2.92·31-s + 32-s − 7.46·34-s − 6.69·37-s − 38-s − 6.69·41-s − 1.79·43-s + 3.86·44-s + 1.46·46-s − 9.46·47-s − 5.92·49-s + 1.03·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.391·7-s + 0.353·8-s + 1.16·11-s + 0.287·13-s − 0.276·14-s + 0.250·16-s − 1.81·17-s − 0.229·19-s + 0.823·22-s + 0.305·23-s + 0.203·26-s − 0.195·28-s − 1.76·29-s + 0.525·31-s + 0.176·32-s − 1.28·34-s − 1.10·37-s − 0.162·38-s − 1.04·41-s − 0.273·43-s + 0.582·44-s + 0.215·46-s − 1.38·47-s − 0.846·49-s + 0.143·52-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 | \( 1 + 1.03T + 7T^{2} \) |
| 11 | \( 1 - 3.86T + 11T^{2} \) |
| 13 | \( 1 - 1.03T + 13T^{2} \) |
| 17 | \( 1 + 7.46T + 17T^{2} \) |
| 23 | \( 1 - 1.46T + 23T^{2} \) |
| 29 | \( 1 + 9.52T + 29T^{2} \) |
| 31 | \( 1 - 2.92T + 31T^{2} \) |
| 37 | \( 1 + 6.69T + 37T^{2} \) |
| 41 | \( 1 + 6.69T + 41T^{2} \) |
| 43 | \( 1 + 1.79T + 43T^{2} \) |
| 47 | \( 1 + 9.46T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 12.6T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 3.58T + 67T^{2} \) |
| 71 | \( 1 - 15.4T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 4.39T + 83T^{2} \) |
| 89 | \( 1 + 1.03T + 89T^{2} \) |
| 97 | \( 1 - 17.2T + 97T^{2} \) |
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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
−7.06803821910788384113658907932, −6.67045166744698232027563618257, −6.21144673719911554374151535173, −5.26636700491076218577964732557, −4.59711051783282614574722854794, −3.81895567213535687052661989691, −3.33949799656411431151792646017, −2.21257853026140678874406563119, −1.52380750299865813559141647063, 0,
1.52380750299865813559141647063, 2.21257853026140678874406563119, 3.33949799656411431151792646017, 3.81895567213535687052661989691, 4.59711051783282614574722854794, 5.26636700491076218577964732557, 6.21144673719911554374151535173, 6.67045166744698232027563618257, 7.06803821910788384113658907932
