| L(s) = 1 | − 9-s + 8·19-s − 4·29-s − 8·31-s + 4·41-s + 14·49-s − 16·59-s + 4·61-s − 16·71-s − 24·79-s + 81-s + 28·89-s + 20·101-s − 20·109-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 + ⋯ |
| L(s) = 1 | − 1/3·9-s + 1.83·19-s − 0.742·29-s − 1.43·31-s + 0.624·41-s + 2·49-s − 2.08·59-s + 0.512·61-s − 1.89·71-s − 2.70·79-s + 1/9·81-s + 2.96·89-s + 1.99·101-s − 1.91·109-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.440051631\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.440051631\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−8.810693558670117024080883748727, −7.78836093625334776465573551175, −7.73883503057465455724837766867, −7.34554876111416156679978408542, −7.34548609825308846561796589929, −6.60784960757236337997500695256, −6.25449061157180928154606929417, −5.82004934949027835210529867676, −5.63527264948320825461346414328, −5.08651605796836415823035017968, −4.97780553932737593400022108817, −4.33034270936793254315718176937, −3.79719581801783884373091038081, −3.66291170833758050763207099438, −3.09088517238200020554199613429, −2.62711201163926188229327463379, −2.31624474995359990651049566964, −1.36201600088555421932110646520, −1.36090735493308643597599855026, −0.33542539082138709577253890089,
0.33542539082138709577253890089, 1.36090735493308643597599855026, 1.36201600088555421932110646520, 2.31624474995359990651049566964, 2.62711201163926188229327463379, 3.09088517238200020554199613429, 3.66291170833758050763207099438, 3.79719581801783884373091038081, 4.33034270936793254315718176937, 4.97780553932737593400022108817, 5.08651605796836415823035017968, 5.63527264948320825461346414328, 5.82004934949027835210529867676, 6.25449061157180928154606929417, 6.60784960757236337997500695256, 7.34548609825308846561796589929, 7.34554876111416156679978408542, 7.73883503057465455724837766867, 7.78836093625334776465573551175, 8.810693558670117024080883748727
