| L(s) = 1 | + 3-s − 4·5-s − 9-s + 2·11-s + 5·13-s − 4·15-s − 2·17-s + 2·19-s + 2·25-s + 3·29-s + 9·31-s + 2·33-s + 5·39-s + 41-s − 16·43-s + 4·45-s − 11·47-s − 14·49-s − 2·51-s − 4·53-s − 8·55-s + 2·57-s − 4·59-s − 8·61-s − 20·65-s − 2·67-s − 23·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s − 1/3·9-s + 0.603·11-s + 1.38·13-s − 1.03·15-s − 0.485·17-s + 0.458·19-s + 2/5·25-s + 0.557·29-s + 1.61·31-s + 0.348·33-s + 0.800·39-s + 0.156·41-s − 2.43·43-s + 0.596·45-s − 1.60·47-s − 2·49-s − 0.280·51-s − 0.549·53-s − 1.07·55-s + 0.264·57-s − 0.520·59-s − 1.02·61-s − 2.48·65-s − 0.244·67-s − 2.72·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71639296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71639296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2577644958\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2577644958\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 23 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 5 T + 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T + 56 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 9 T + 78 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - T - 24 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 11 T + 86 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 118 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 23 T + 270 T^{2} + 23 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 17 T + 180 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 142 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 26 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−7.85838944083928628799149735806, −7.85659029829258928502662653567, −7.44210815146827147767737038397, −6.89679656369428960862380703837, −6.51664671771786227175236908413, −6.46757672478905656516308767860, −5.93053967677957964968332954843, −5.69937149134438649691768488819, −4.90028860211417688318632747425, −4.62141168392115720107673106744, −4.51960436590143837754952749105, −4.03890874971068456714478060464, −3.50838920432675610128730631649, −3.34589865129688811718225709171, −3.05356403403529557848582219993, −2.73978509636157557045997400230, −1.68758634992823606684721643363, −1.66225372085762921514764712797, −1.01392408825779667377518916725, −0.12511440987640091051321498059,
0.12511440987640091051321498059, 1.01392408825779667377518916725, 1.66225372085762921514764712797, 1.68758634992823606684721643363, 2.73978509636157557045997400230, 3.05356403403529557848582219993, 3.34589865129688811718225709171, 3.50838920432675610128730631649, 4.03890874971068456714478060464, 4.51960436590143837754952749105, 4.62141168392115720107673106744, 4.90028860211417688318632747425, 5.69937149134438649691768488819, 5.93053967677957964968332954843, 6.46757672478905656516308767860, 6.51664671771786227175236908413, 6.89679656369428960862380703837, 7.44210815146827147767737038397, 7.85659029829258928502662653567, 7.85838944083928628799149735806
