| L(s) = 1 | + 2-s + 4-s + 5·7-s + 8-s − 5·9-s + 5·14-s + 16-s + 3·17-s − 5·18-s + 7·23-s − 6·25-s + 5·28-s + 10·31-s + 32-s + 3·34-s − 5·36-s − 8·41-s + 7·46-s + 9·47-s + 7·49-s − 6·50-s + 5·56-s + 10·62-s − 25·63-s + 64-s + 3·68-s + 3·71-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.88·7-s + 0.353·8-s − 5/3·9-s + 1.33·14-s + 1/4·16-s + 0.727·17-s − 1.17·18-s + 1.45·23-s − 6/5·25-s + 0.944·28-s + 1.79·31-s + 0.176·32-s + 0.514·34-s − 5/6·36-s − 1.24·41-s + 1.03·46-s + 1.31·47-s + 49-s − 0.848·50-s + 0.668·56-s + 1.27·62-s − 3.14·63-s + 1/8·64-s + 0.363·68-s + 0.356·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.813574304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.813574304\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 8 T + p T^{2} ) \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 9 T + p T^{2} ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 43 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 41 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
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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
−9.348332074657135541786375582554, −8.814112526295927591015416473931, −8.289198058126817711835012991516, −8.006948815955006424557163336220, −7.66410544265335292360152139133, −6.85259628036323645450189122355, −6.35840524133394943502182258010, −5.60075179353449170973654143302, −5.36331340210760167215393837541, −4.87195598885244445538802671871, −4.31021775389373398970788807990, −3.48273957978546659479591191735, −2.83941599262777214057979742016, −2.17299843448316856009083727684, −1.18306152443171949635429869335,
1.18306152443171949635429869335, 2.17299843448316856009083727684, 2.83941599262777214057979742016, 3.48273957978546659479591191735, 4.31021775389373398970788807990, 4.87195598885244445538802671871, 5.36331340210760167215393837541, 5.60075179353449170973654143302, 6.35840524133394943502182258010, 6.85259628036323645450189122355, 7.66410544265335292360152139133, 8.006948815955006424557163336220, 8.289198058126817711835012991516, 8.814112526295927591015416473931, 9.348332074657135541786375582554
