| L(s) = 1 | + 4-s − 3·5-s + 5·9-s + 16-s + 16·19-s − 3·20-s + 5·25-s + 12·29-s + 2·31-s + 5·36-s − 12·41-s − 15·45-s + 4·49-s + 18·59-s − 20·61-s + 5·64-s − 6·71-s + 16·76-s + 28·79-s − 3·80-s + 9·81-s + 6·89-s − 48·95-s + 5·100-s − 24·101-s + 40·109-s + 12·116-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 1.34·5-s + 5/3·9-s + 1/4·16-s + 3.67·19-s − 0.670·20-s + 25-s + 2.22·29-s + 0.359·31-s + 5/6·36-s − 1.87·41-s − 2.23·45-s + 4/7·49-s + 2.34·59-s − 2.56·61-s + 5/8·64-s − 0.712·71-s + 1.83·76-s + 3.15·79-s − 0.335·80-s + 81-s + 0.635·89-s − 4.92·95-s + 1/2·100-s − 2.38·101-s + 3.83·109-s + 1.11·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.826656504\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.826656504\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - T^{2} - p^{2} T^{6} + p^{4} T^{8} \) |
| 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} )( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 846 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 - 85 T^{2} + 2856 T^{4} - 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - T + 54 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} ) \) |
| 41 | $D_{4}$ | \( ( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 6006 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 9 T + 130 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 10 T + 114 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 181 T^{2} + 15312 T^{4} - 181 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 3 T + 70 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 14 T + 174 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 3 T + 172 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 337 T^{2} + 47136 T^{4} - 337 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.62312346452348710948298320952, −7.56167110682935819631897649123, −7.18655902884916164388664087066, −6.91189455124613568041877591161, −6.86155895943070410537506426622, −6.62839621550305437245631857300, −6.51749051782981139691969967219, −5.91036376415282172734571149765, −5.76747495233482591773060691658, −5.42657400093018570778408055405, −5.21174472340115466090675591112, −4.86680725215035031316690321782, −4.68381839900749743337294279008, −4.57078960261336057308232382301, −4.14738754503175398895066481644, −3.82449024088773432808378203551, −3.47675716629220954232490730031, −3.28458531625869575087816792307, −3.24076928710729858462834979968, −2.62445275935766940188297970435, −2.48924736349072704104789548393, −1.75634307146198927034850509194, −1.46807841895572030158506165825, −0.928349196478761223245163912915, −0.78589337587497959640231614419,
0.78589337587497959640231614419, 0.928349196478761223245163912915, 1.46807841895572030158506165825, 1.75634307146198927034850509194, 2.48924736349072704104789548393, 2.62445275935766940188297970435, 3.24076928710729858462834979968, 3.28458531625869575087816792307, 3.47675716629220954232490730031, 3.82449024088773432808378203551, 4.14738754503175398895066481644, 4.57078960261336057308232382301, 4.68381839900749743337294279008, 4.86680725215035031316690321782, 5.21174472340115466090675591112, 5.42657400093018570778408055405, 5.76747495233482591773060691658, 5.91036376415282172734571149765, 6.51749051782981139691969967219, 6.62839621550305437245631857300, 6.86155895943070410537506426622, 6.91189455124613568041877591161, 7.18655902884916164388664087066, 7.56167110682935819631897649123, 7.62312346452348710948298320952
