| L(s) = 1 | − 8·4-s + 9·5-s + 6·11-s + 48·16-s − 38·19-s − 72·20-s + 56·25-s − 48·44-s − 73·49-s + 54·55-s − 206·61-s − 256·64-s + 304·76-s + 432·80-s − 342·95-s − 448·100-s − 204·101-s − 215·121-s + 279·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 2·4-s + 9/5·5-s + 6/11·11-s + 3·16-s − 2·19-s − 3.59·20-s + 2.23·25-s − 1.09·44-s − 1.48·49-s + 0.981·55-s − 3.37·61-s − 4·64-s + 4·76-s + 27/5·80-s − 3.59·95-s − 4.47·100-s − 2.01·101-s − 1.77·121-s + 2.23·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.346109217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.346109217\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 9 T + p^{2} T^{2} \) |
| 19 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p^{2} T^{2} )( 1 + 5 T + p^{2} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 15 T + p^{2} T^{2} )( 1 + 15 T + p^{2} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 30 T + p^{2} T^{2} )( 1 + 30 T + p^{2} T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 85 T + p^{2} T^{2} )( 1 + 85 T + p^{2} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 75 T + p^{2} T^{2} )( 1 + 75 T + p^{2} T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 103 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 25 T + p^{2} T^{2} )( 1 + 25 T + p^{2} T^{2} ) \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 90 T + p^{2} T^{2} )( 1 + 90 T + p^{2} T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
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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
−10.21411103908636647654581436197, −9.550842468510577781871009122963, −9.400871792025640174763422465049, −9.073566620157399150618481857974, −8.614997221610750450508404222912, −8.322030024886761074478638945232, −7.86535701289578131786832250463, −7.15558748746560826001833132181, −6.51995805698792471569870531927, −6.06133435366210021883234367630, −5.97240980051552956224109535402, −5.20346406440331854273489059783, −4.92132482015654401759069009510, −4.37262569318665107873348789605, −4.07599810484994994673126727434, −3.27285391371655388459377380395, −2.74291947162393035842154517126, −1.76823942707055177821230305045, −1.43959609381118701976508448487, −0.40827917194358246516163146790,
0.40827917194358246516163146790, 1.43959609381118701976508448487, 1.76823942707055177821230305045, 2.74291947162393035842154517126, 3.27285391371655388459377380395, 4.07599810484994994673126727434, 4.37262569318665107873348789605, 4.92132482015654401759069009510, 5.20346406440331854273489059783, 5.97240980051552956224109535402, 6.06133435366210021883234367630, 6.51995805698792471569870531927, 7.15558748746560826001833132181, 7.86535701289578131786832250463, 8.322030024886761074478638945232, 8.614997221610750450508404222912, 9.073566620157399150618481857974, 9.400871792025640174763422465049, 9.550842468510577781871009122963, 10.21411103908636647654581436197
