L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 2·11-s + 6·13-s − 14-s + 16-s − 6·17-s − 2·22-s + 23-s + 6·26-s − 28-s + 4·29-s − 2·31-s + 32-s − 6·34-s − 4·37-s + 2·41-s + 4·43-s − 2·44-s + 46-s + 49-s + 6·52-s + 6·53-s − 56-s + 4·58-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 0.603·11-s + 1.66·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.426·22-s + 0.208·23-s + 1.17·26-s − 0.188·28-s + 0.742·29-s − 0.359·31-s + 0.176·32-s − 1.02·34-s − 0.657·37-s + 0.312·41-s + 0.609·43-s − 0.301·44-s + 0.147·46-s + 1/7·49-s + 0.832·52-s + 0.824·53-s − 0.133·56-s + 0.525·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.15186229328985, −13.75821866695076, −13.37149536520961, −12.92913821330352, −12.58371669647273, −11.81800737924831, −11.35963378393646, −10.91961139711363, −10.45585883422021, −10.02359140106335, −9.088495710424194, −8.766236883452666, −8.336966499157762, −7.510725627229990, −7.076203598878625, −6.379822425157138, −6.117288612829042, −5.468403759365634, −4.873399914590519, −4.157439293998762, −3.840340879129332, −3.031180919426583, −2.572874712606028, −1.782949928213085, −1.031096301372602, 0,
1.031096301372602, 1.782949928213085, 2.572874712606028, 3.031180919426583, 3.840340879129332, 4.157439293998762, 4.873399914590519, 5.468403759365634, 6.117288612829042, 6.379822425157138, 7.076203598878625, 7.510725627229990, 8.336966499157762, 8.766236883452666, 9.088495710424194, 10.02359140106335, 10.45585883422021, 10.91961139711363, 11.35963378393646, 11.81800737924831, 12.58371669647273, 12.92913821330352, 13.37149536520961, 13.75821866695076, 14.15186229328985