| L(s) = 1 | − 8·11-s + 10·19-s − 4·29-s + 14·31-s + 10·49-s + 28·59-s − 22·61-s + 6·79-s − 24·101-s + 38·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 2.41·11-s + 2.29·19-s − 0.742·29-s + 2.51·31-s + 10/7·49-s + 3.64·59-s − 2.81·61-s + 0.675·79-s − 2.38·101-s + 3.63·109-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 + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.472537162\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.472537162\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 165 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−8.345270183361947044599939969315, −7.975715896203517819245614427972, −7.51341788346161165351386086067, −7.44615503305948131248514102896, −7.10789473179686956302803286314, −6.62069475559202915648586693719, −5.97811384516288082467007100774, −5.88320244115332881198355048646, −5.38505375582839570961371897164, −5.11400302517237359093259604316, −4.84453364938573235203482031350, −4.42867012941824790532438985113, −3.80465457221140442853936743674, −3.45089141863871636893810066595, −2.81095020440496331154088398882, −2.75859437368975164278226958323, −2.32446047100163260834764846634, −1.62541473600245785047055149123, −0.902818176387185737754311669214, −0.51320670619519088892062977641,
0.51320670619519088892062977641, 0.902818176387185737754311669214, 1.62541473600245785047055149123, 2.32446047100163260834764846634, 2.75859437368975164278226958323, 2.81095020440496331154088398882, 3.45089141863871636893810066595, 3.80465457221140442853936743674, 4.42867012941824790532438985113, 4.84453364938573235203482031350, 5.11400302517237359093259604316, 5.38505375582839570961371897164, 5.88320244115332881198355048646, 5.97811384516288082467007100774, 6.62069475559202915648586693719, 7.10789473179686956302803286314, 7.44615503305948131248514102896, 7.51341788346161165351386086067, 7.975715896203517819245614427972, 8.345270183361947044599939969315
