| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 3·9-s − 4·11-s − 14-s + 16-s − 2·17-s − 3·18-s − 4·22-s − 28-s + 6·29-s − 8·31-s + 32-s − 2·34-s − 3·36-s − 10·37-s − 2·41-s − 4·43-s − 4·44-s + 8·47-s + 49-s + 2·53-s − 56-s + 6·58-s + 8·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 9-s − 1.20·11-s − 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.707·18-s − 0.852·22-s − 0.188·28-s + 1.11·29-s − 1.43·31-s + 0.176·32-s − 0.342·34-s − 1/2·36-s − 1.64·37-s − 0.312·41-s − 0.609·43-s − 0.603·44-s + 1.16·47-s + 1/7·49-s + 0.274·53-s − 0.133·56-s + 0.787·58-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.171189890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.171189890\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−14.20095687594553, −13.71217946323455, −13.48300649915085, −12.85228885562701, −12.17351807879080, −12.11642549393581, −11.23064794454889, −10.78290681777153, −10.47446249824106, −9.825057693388684, −9.053365148732477, −8.655559310173651, −8.055473528558796, −7.501914796856152, −6.842637626106368, −6.432219058261266, −5.579124051254289, −5.408803327247201, −4.799926476431122, −4.012886902871024, −3.401267441007611, −2.799427949849874, −2.366835609923972, −1.528529810616742, −0.3071963739223118,
0.3071963739223118, 1.528529810616742, 2.366835609923972, 2.799427949849874, 3.401267441007611, 4.012886902871024, 4.799926476431122, 5.408803327247201, 5.579124051254289, 6.432219058261266, 6.842637626106368, 7.501914796856152, 8.055473528558796, 8.655559310173651, 9.053365148732477, 9.825057693388684, 10.47446249824106, 10.78290681777153, 11.23064794454889, 12.11642549393581, 12.17351807879080, 12.85228885562701, 13.48300649915085, 13.71217946323455, 14.20095687594553
