| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 13-s + 14-s + 16-s − 3·17-s + 4·19-s − 3·23-s − 26-s + 28-s + 3·29-s + 5·31-s + 32-s − 3·34-s + 10·37-s + 4·38-s + 9·41-s − 43-s − 3·46-s + 49-s − 52-s + 9·53-s + 56-s + 3·58-s − 9·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.917·19-s − 0.625·23-s − 0.196·26-s + 0.188·28-s + 0.557·29-s + 0.898·31-s + 0.176·32-s − 0.514·34-s + 1.64·37-s + 0.648·38-s + 1.40·41-s − 0.152·43-s − 0.442·46-s + 1/7·49-s − 0.138·52-s + 1.23·53-s + 0.133·56-s + 0.393·58-s − 1.17·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 381150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.60113083422119, −12.14487842238803, −12.02336929180955, −11.34617665529785, −10.98766482538537, −10.63765435082784, −10.00621949902745, −9.478707124562973, −9.276464628526203, −8.451144334191333, −8.058887433008217, −7.653106602900445, −7.205774711798738, −6.485078081406124, −6.339182649662990, −5.584280366526552, −5.309534182790650, −4.613872313990015, −4.251825819897504, −3.916971867489447, −3.020396553915879, −2.666334283655824, −2.238463741167533, −1.379403539985903, −0.9557618517437279, 0,
0.9557618517437279, 1.379403539985903, 2.238463741167533, 2.666334283655824, 3.020396553915879, 3.916971867489447, 4.251825819897504, 4.613872313990015, 5.309534182790650, 5.584280366526552, 6.339182649662990, 6.485078081406124, 7.205774711798738, 7.653106602900445, 8.058887433008217, 8.451144334191333, 9.276464628526203, 9.478707124562973, 10.00621949902745, 10.63765435082784, 10.98766482538537, 11.34617665529785, 12.02336929180955, 12.14487842238803, 12.60113083422119
