| L(s) = 1 | + 2·9-s + 2·11-s − 8·23-s − 8·25-s − 16·37-s − 8·43-s − 12·53-s + 16·67-s − 16·71-s − 32·79-s − 5·81-s + 4·99-s + 16·107-s + 32·109-s + 12·113-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 24·169-s + 173-s + ⋯ |
| L(s) = 1 | + 2/3·9-s + 0.603·11-s − 1.66·23-s − 8/5·25-s − 2.63·37-s − 1.21·43-s − 1.64·53-s + 1.95·67-s − 1.89·71-s − 3.60·79-s − 5/9·81-s + 0.402·99-s + 1.54·107-s + 3.06·109-s + 1.12·113-s + 3/11·121-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 − 1.84·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74373376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74373376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 120 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 144 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 158 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 64 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 96 T^{2} + p^{2} T^{4} \) |
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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
−7.46760761991361247988733304208, −7.29539316156711661568897171555, −7.04413911466501462583926811631, −6.47726598872353544502337158923, −6.26009793297953143686702153148, −5.95446841228358925288734060977, −5.56911407749023556267444126448, −5.17855580284254959492776254623, −4.79348895076068348219361886611, −4.29657086124467134879326308352, −4.13480041904322698132155791797, −3.70980917516283042942945851155, −3.23965352148593878850875245520, −3.10662169369221883707936802326, −2.18229187698172178580742072031, −1.91441368626081719640283111893, −1.64898519759039438656792173194, −1.12113155743030734812238244327, 0, 0,
1.12113155743030734812238244327, 1.64898519759039438656792173194, 1.91441368626081719640283111893, 2.18229187698172178580742072031, 3.10662169369221883707936802326, 3.23965352148593878850875245520, 3.70980917516283042942945851155, 4.13480041904322698132155791797, 4.29657086124467134879326308352, 4.79348895076068348219361886611, 5.17855580284254959492776254623, 5.56911407749023556267444126448, 5.95446841228358925288734060977, 6.26009793297953143686702153148, 6.47726598872353544502337158923, 7.04413911466501462583926811631, 7.29539316156711661568897171555, 7.46760761991361247988733304208
