| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 4·11-s − 14-s + 16-s + 6·17-s + 6·19-s + 4·22-s − 5·25-s − 28-s − 10·29-s + 4·31-s + 32-s + 6·34-s + 2·37-s + 6·38-s + 10·41-s + 4·43-s + 4·44-s − 12·47-s + 49-s − 5·50-s − 6·53-s − 56-s − 10·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 1.20·11-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 1.37·19-s + 0.852·22-s − 25-s − 0.188·28-s − 1.85·29-s + 0.718·31-s + 0.176·32-s + 1.02·34-s + 0.328·37-s + 0.973·38-s + 1.56·41-s + 0.609·43-s + 0.603·44-s − 1.75·47-s + 1/7·49-s − 0.707·50-s − 0.824·53-s − 0.133·56-s − 1.31·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66654 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66654 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.922119754\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.922119754\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−14.16630693058600, −13.89287250930063, −13.04338640670130, −12.83955270482338, −12.12405337655661, −11.77680955922478, −11.32674785647722, −10.89019190680716, −9.876561937850978, −9.701594262131962, −9.358160726802383, −8.493848657513707, −7.763406982273617, −7.488569814428203, −6.889027052728525, −6.160550603266127, −5.781846148282814, −5.360918488507144, −4.509677399679625, −3.990897492296087, −3.351754002053125, −3.083218449226977, −2.049374877074225, −1.406293551129801, −0.6915699768351328,
0.6915699768351328, 1.406293551129801, 2.049374877074225, 3.083218449226977, 3.351754002053125, 3.990897492296087, 4.509677399679625, 5.360918488507144, 5.781846148282814, 6.160550603266127, 6.889027052728525, 7.488569814428203, 7.763406982273617, 8.493848657513707, 9.358160726802383, 9.701594262131962, 9.876561937850978, 10.89019190680716, 11.32674785647722, 11.77680955922478, 12.12405337655661, 12.83955270482338, 13.04338640670130, 13.89287250930063, 14.16630693058600
