| L(s) = 1 | − 2·4-s − 2·9-s + 4·16-s − 10·25-s + 4·36-s + 20·43-s − 7·49-s − 8·64-s − 5·81-s + 20·100-s + 12·107-s − 36·113-s + 14·121-s + 127-s + 131-s + 137-s + 139-s − 8·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 13·169-s − 40·172-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 4-s − 2/3·9-s + 16-s − 2·25-s + 2/3·36-s + 3.04·43-s − 49-s − 64-s − 5/9·81-s + 2·100-s + 1.16·107-s − 3.38·113-s + 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 169-s − 3.04·172-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9406444143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9406444143\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| 13 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 158 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 146 T^{2} + p^{2} T^{4} \) |
| 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
−8.508960193822939486478690692279, −8.045955124583760802852609303993, −7.62996356347171534713933850235, −7.38480514001354382232095887813, −6.49116829918350661385933701592, −6.12234244835669884186306396683, −5.55028250842661352481992762397, −5.41101012351539568677522043386, −4.59719182905216380872528147414, −4.14699901367412754328218294373, −3.77676223329212239240206310018, −3.05871939896826161251031442694, −2.44878784497011915880737547029, −1.59626055961315407632144667284, −0.51837991524260537333013625572,
0.51837991524260537333013625572, 1.59626055961315407632144667284, 2.44878784497011915880737547029, 3.05871939896826161251031442694, 3.77676223329212239240206310018, 4.14699901367412754328218294373, 4.59719182905216380872528147414, 5.41101012351539568677522043386, 5.55028250842661352481992762397, 6.12234244835669884186306396683, 6.49116829918350661385933701592, 7.38480514001354382232095887813, 7.62996356347171534713933850235, 8.045955124583760802852609303993, 8.508960193822939486478690692279
