| L(s) = 1 | + 2·9-s + 8·11-s + 16·19-s − 4·29-s + 8·31-s − 20·41-s + 10·49-s − 4·61-s − 24·71-s + 16·79-s − 5·81-s + 12·89-s + 16·99-s − 28·101-s − 28·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 32·171-s + ⋯ |
| L(s) = 1 | + 2/3·9-s + 2.41·11-s + 3.67·19-s − 0.742·29-s + 1.43·31-s − 3.12·41-s + 10/7·49-s − 0.512·61-s − 2.84·71-s + 1.80·79-s − 5/9·81-s + 1.27·89-s + 1.60·99-s − 2.78·101-s − 2.68·109-s + 2.36·121-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 − 0.769·169-s + 2.44·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.627779276\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.627779276\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−9.403920090483631738283990942456, −9.256500221090967750008285562604, −9.117952306469587796120860827362, −8.384984644918352068433634988480, −8.024469776857550326543995507809, −7.53369881751463538510417247131, −7.08236231388169379700618667054, −6.92401728951112181007223304880, −6.47903964811604642931322440319, −6.01351032673423886002401589310, −5.36373816776797327755562939933, −5.21358882191819353482594271906, −4.60215591635517243512643127700, −3.89658893442426124530836401212, −3.86411930201301308374933655949, −3.10586961599280682809307628668, −2.88629165733295320274301106504, −1.56454295220466950950184753493, −1.50774740192706163630300199086, −0.850242754109977132122766858480,
0.850242754109977132122766858480, 1.50774740192706163630300199086, 1.56454295220466950950184753493, 2.88629165733295320274301106504, 3.10586961599280682809307628668, 3.86411930201301308374933655949, 3.89658893442426124530836401212, 4.60215591635517243512643127700, 5.21358882191819353482594271906, 5.36373816776797327755562939933, 6.01351032673423886002401589310, 6.47903964811604642931322440319, 6.92401728951112181007223304880, 7.08236231388169379700618667054, 7.53369881751463538510417247131, 8.024469776857550326543995507809, 8.384984644918352068433634988480, 9.117952306469587796120860827362, 9.256500221090967750008285562604, 9.403920090483631738283990942456
