| L(s) = 1 | + 3·3-s + 4-s − 2·7-s + 2·9-s − 11-s + 3·12-s − 13-s − 3·16-s + 17-s − 13·19-s − 6·21-s − 6·27-s − 2·28-s − 3·29-s − 4·31-s − 3·33-s + 2·36-s − 3·39-s + 7·41-s + 12·43-s − 44-s − 9·47-s − 9·48-s − 6·49-s + 3·51-s − 52-s − 12·53-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1/2·4-s − 0.755·7-s + 2/3·9-s − 0.301·11-s + 0.866·12-s − 0.277·13-s − 3/4·16-s + 0.242·17-s − 2.98·19-s − 1.30·21-s − 1.15·27-s − 0.377·28-s − 0.557·29-s − 0.718·31-s − 0.522·33-s + 1/3·36-s − 0.480·39-s + 1.09·41-s + 1.82·43-s − 0.150·44-s − 1.31·47-s − 1.29·48-s − 6/7·49-s + 0.420·51-s − 0.138·52-s − 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36300625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36300625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 241 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - p T + 7 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 + T + 21 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + T + 15 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $C_4$ | \( 1 + 13 T + 79 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 7 T + 93 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 9 T + 113 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T + 74 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_4$ | \( 1 - 19 T + 201 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 3 T + 105 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 5 T + 47 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 13 T + 187 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 13 T + 177 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_4$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 10 T + 214 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.72023569051927546319675957962, −7.65554017003726042078808783941, −7.51075775172702162864921769887, −6.71097192513621731480339720349, −6.57362372753031299236837580019, −6.22756360917029974626910395028, −5.96951053508374865079595143693, −5.43568323635522105852349836635, −4.83764846851672347112498594709, −4.61307361893413625673697890803, −4.00555827203506744141800103166, −3.68389237844895385228866595873, −3.49750685419370791734127483385, −2.81903563313289733415372678894, −2.42387719408229139623850279187, −2.37448751809789495132666278067, −1.97806905444663235744401182361, −1.27369652867433214428229577910, 0, 0,
1.27369652867433214428229577910, 1.97806905444663235744401182361, 2.37448751809789495132666278067, 2.42387719408229139623850279187, 2.81903563313289733415372678894, 3.49750685419370791734127483385, 3.68389237844895385228866595873, 4.00555827203506744141800103166, 4.61307361893413625673697890803, 4.83764846851672347112498594709, 5.43568323635522105852349836635, 5.96951053508374865079595143693, 6.22756360917029974626910395028, 6.57362372753031299236837580019, 6.71097192513621731480339720349, 7.51075775172702162864921769887, 7.65554017003726042078808783941, 7.72023569051927546319675957962
