| L(s) = 1 | − 5-s − 8·11-s + 13-s + 3·17-s + 8·19-s − 5·23-s + 5·25-s + 7·29-s − 8·31-s + 20·37-s − 5·41-s − 5·43-s + 7·47-s − 14·49-s + 11·53-s + 8·55-s − 3·59-s − 11·61-s − 65-s − 3·67-s − 11·71-s − 15·73-s − 13·79-s − 3·85-s + 3·89-s − 8·95-s + 5·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 2.41·11-s + 0.277·13-s + 0.727·17-s + 1.83·19-s − 1.04·23-s + 25-s + 1.29·29-s − 1.43·31-s + 3.28·37-s − 0.780·41-s − 0.762·43-s + 1.02·47-s − 2·49-s + 1.51·53-s + 1.07·55-s − 0.390·59-s − 1.40·61-s − 0.124·65-s − 0.366·67-s − 1.30·71-s − 1.75·73-s − 1.46·79-s − 0.325·85-s + 0.317·89-s − 0.820·95-s + 0.507·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.359144307\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.359144307\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 5 T + 2 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 7 T + 20 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 7 T + 2 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 11 T + 68 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 11 T + 50 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 15 T + 152 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.054705403434683266982623551208, −8.466716316120410172825665480837, −8.140789053064287855912961104240, −7.87920015256553347887430123904, −7.52076972315539384232848272679, −7.36794017330594851920613005949, −6.88361181659730053854982346490, −6.13773872994092676326782427873, −5.92280105664678155302966185313, −5.55715079039399319366086991756, −5.11424772070726881081185604944, −4.75054825821794167386846614079, −4.39502527415872780641166664417, −3.80394392294044327547846539700, −3.12014877904489795241903560323, −2.91629299416387880212151279329, −2.67666825197449309790647824329, −1.79115706186670023341258478109, −1.15918922571281169518201343187, −0.41216880717598192294647371678,
0.41216880717598192294647371678, 1.15918922571281169518201343187, 1.79115706186670023341258478109, 2.67666825197449309790647824329, 2.91629299416387880212151279329, 3.12014877904489795241903560323, 3.80394392294044327547846539700, 4.39502527415872780641166664417, 4.75054825821794167386846614079, 5.11424772070726881081185604944, 5.55715079039399319366086991756, 5.92280105664678155302966185313, 6.13773872994092676326782427873, 6.88361181659730053854982346490, 7.36794017330594851920613005949, 7.52076972315539384232848272679, 7.87920015256553347887430123904, 8.140789053064287855912961104240, 8.466716316120410172825665480837, 9.054705403434683266982623551208
