| L(s) = 1 | − 12·11-s + 2·19-s + 12·29-s − 6·31-s − 8·41-s − 11·49-s + 12·59-s + 6·61-s − 24·71-s − 16·79-s − 32·89-s + 16·101-s + 14·109-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 17·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 3.61·11-s + 0.458·19-s + 2.22·29-s − 1.07·31-s − 1.24·41-s − 1.57·49-s + 1.56·59-s + 0.768·61-s − 2.84·71-s − 1.80·79-s − 3.39·89-s + 1.59·101-s + 1.34·109-s + 7.81·121-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 + 1.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2422954643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2422954643\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 133 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
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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
−8.761829373518809644484441660582, −8.418497577922149638990554646212, −7.978311880937303711520830107156, −7.62210323650550110214567373342, −7.41023975045528829507494880991, −6.90691187074294192910110379441, −6.68313131548640812923679635311, −5.80040216036889558361025138667, −5.79459329458594167822414392923, −5.34795891099135184522422890147, −4.98332307613921905042812395949, −4.60954262544018059563326706129, −4.36746134032781799124493500280, −3.32331425770543258610521642348, −3.29475390908325167088891607720, −2.63387254527647022482638134530, −2.52191910987588361070693024494, −1.84165935382633995903437928862, −1.12272263951553656008157393063, −0.15185495354903277295640161321,
0.15185495354903277295640161321, 1.12272263951553656008157393063, 1.84165935382633995903437928862, 2.52191910987588361070693024494, 2.63387254527647022482638134530, 3.29475390908325167088891607720, 3.32331425770543258610521642348, 4.36746134032781799124493500280, 4.60954262544018059563326706129, 4.98332307613921905042812395949, 5.34795891099135184522422890147, 5.79459329458594167822414392923, 5.80040216036889558361025138667, 6.68313131548640812923679635311, 6.90691187074294192910110379441, 7.41023975045528829507494880991, 7.62210323650550110214567373342, 7.978311880937303711520830107156, 8.418497577922149638990554646212, 8.761829373518809644484441660582
