| L(s) = 1 | + 3·7-s − 9-s − 2·11-s + 9·13-s + 6·17-s + 13·19-s + 2·23-s + 11·29-s + 4·31-s − 12·37-s − 2·41-s + 11·43-s + 7·47-s − 6·49-s − 2·53-s + 7·59-s − 4·61-s − 3·63-s + 14·67-s − 15·71-s + 22·73-s − 6·77-s + 5·79-s − 8·81-s − 18·83-s + 8·89-s + 27·91-s + ⋯ |
| L(s) = 1 | + 1.13·7-s − 1/3·9-s − 0.603·11-s + 2.49·13-s + 1.45·17-s + 2.98·19-s + 0.417·23-s + 2.04·29-s + 0.718·31-s − 1.97·37-s − 0.312·41-s + 1.67·43-s + 1.02·47-s − 6/7·49-s − 0.274·53-s + 0.911·59-s − 0.512·61-s − 0.377·63-s + 1.71·67-s − 1.78·71-s + 2.57·73-s − 0.683·77-s + 0.562·79-s − 8/9·81-s − 1.97·83-s + 0.847·89-s + 2.83·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25000000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25000000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.105334261\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.105334261\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 3 T + 15 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 9 T + 45 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_4$ | \( 1 - 13 T + 79 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 11 T + 3 p T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 105 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 38 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 11 T + 115 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 7 T + 95 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 7 T + 129 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 14 T + 138 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 15 T + 197 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 5 T + 103 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 3 T - 85 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
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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
−8.192660890612915937822827089075, −8.171468694813778809144105003464, −7.906789520717673940418250839956, −7.39545509050943832331015684352, −6.93989234591975969654859890725, −6.80169174494682962693753143714, −6.09304979740468563550205765905, −5.82163544502398333233585073069, −5.40321432622224000307388270604, −5.31193522432327444491710009300, −4.85700885409352141610756859134, −4.39466907416120587769116232784, −3.82990876918081890920588906452, −3.45365841968313447796631580677, −2.99440252639100394478674746036, −2.97055251605350105672258357243, −2.07009313886406504464342892506, −1.40296079073813550058691134081, −0.990239120071540947013169629800, −0.925563979991074371201302994640,
0.925563979991074371201302994640, 0.990239120071540947013169629800, 1.40296079073813550058691134081, 2.07009313886406504464342892506, 2.97055251605350105672258357243, 2.99440252639100394478674746036, 3.45365841968313447796631580677, 3.82990876918081890920588906452, 4.39466907416120587769116232784, 4.85700885409352141610756859134, 5.31193522432327444491710009300, 5.40321432622224000307388270604, 5.82163544502398333233585073069, 6.09304979740468563550205765905, 6.80169174494682962693753143714, 6.93989234591975969654859890725, 7.39545509050943832331015684352, 7.906789520717673940418250839956, 8.171468694813778809144105003464, 8.192660890612915937822827089075
