| L(s) = 1 | − 9-s + 8·19-s − 12·29-s + 12·41-s − 49-s + 8·59-s − 4·61-s − 24·71-s − 16·79-s + 81-s − 12·89-s + 4·101-s + 4·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s − 8·171-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 1.83·19-s − 2.22·29-s + 1.87·41-s − 1/7·49-s + 1.04·59-s − 0.512·61-s − 2.84·71-s − 1.80·79-s + 1/9·81-s − 1.27·89-s + 0.398·101-s + 0.383·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s − 0.611·171-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.481248330\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.481248330\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−8.731389190710719312246838151976, −8.127719017188998473047456523531, −7.75568220585696313257809566882, −7.46107571676750112429899395333, −7.30542493285650381387568246867, −6.86415080275670331355426531513, −6.34322366678579608174246648589, −5.81493158278831501214334387892, −5.69539374360081889071383872868, −5.38355340327163034260751658102, −4.92813346992072980744438078575, −4.26745768861729285473387926074, −4.17993366658653661471918241655, −3.41951420391108373278948396551, −3.33211936903594123293663224430, −2.55866283500538369977374090726, −2.45366273357866099928135670919, −1.39885619122335403243424770546, −1.38217103883874459964321539377, −0.35304240272748223640194435041,
0.35304240272748223640194435041, 1.38217103883874459964321539377, 1.39885619122335403243424770546, 2.45366273357866099928135670919, 2.55866283500538369977374090726, 3.33211936903594123293663224430, 3.41951420391108373278948396551, 4.17993366658653661471918241655, 4.26745768861729285473387926074, 4.92813346992072980744438078575, 5.38355340327163034260751658102, 5.69539374360081889071383872868, 5.81493158278831501214334387892, 6.34322366678579608174246648589, 6.86415080275670331355426531513, 7.30542493285650381387568246867, 7.46107571676750112429899395333, 7.75568220585696313257809566882, 8.127719017188998473047456523531, 8.731389190710719312246838151976
