| L(s) = 1 | + 2·3-s + 5-s + 2·7-s + 3·9-s + 4·11-s + 3·13-s + 2·15-s + 2·19-s + 4·21-s − 2·23-s − 8·25-s + 4·27-s + 2·29-s + 8·31-s + 8·33-s + 2·35-s − 2·37-s + 6·39-s + 6·41-s + 5·43-s + 3·45-s + 6·47-s + 3·49-s − 3·53-s + 4·55-s + 4·57-s + 9·59-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s + 0.755·7-s + 9-s + 1.20·11-s + 0.832·13-s + 0.516·15-s + 0.458·19-s + 0.872·21-s − 0.417·23-s − 8/5·25-s + 0.769·27-s + 0.371·29-s + 1.43·31-s + 1.39·33-s + 0.338·35-s − 0.328·37-s + 0.960·39-s + 0.937·41-s + 0.762·43-s + 0.447·45-s + 0.875·47-s + 3/7·49-s − 0.412·53-s + 0.539·55-s + 0.529·57-s + 1.17·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59721984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59721984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.443145146\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.443145146\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 3 T + 27 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 29 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 39 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 73 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 55 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 71 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 5 T - 9 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 77 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 9 T + 137 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T + 67 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 9 T + 153 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 15 T + 187 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 14 T + 115 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 12 T + 189 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 87 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - T + 167 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 193 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.994920884572981285922036248461, −7.88840432579027661822205047037, −7.28517638205992651476170625719, −7.23921495253468541444323295624, −6.52995576064869421005831147608, −6.42284192321717828423830798555, −5.85588904112827243672923891660, −5.85257914777428817189940596320, −5.11041422887724138233122285133, −4.88562457715919204178376484130, −4.23458362790248218519386092577, −4.13932672971655623186674626456, −3.62152119197582334243797673319, −3.55973890271065193943932726121, −2.68545256378962616170519853017, −2.56485909336390661346747036875, −1.97429829735326178112633537233, −1.65397877934486254184356511785, −1.10868062469485658272227241794, −0.73689445958994060240958765981,
0.73689445958994060240958765981, 1.10868062469485658272227241794, 1.65397877934486254184356511785, 1.97429829735326178112633537233, 2.56485909336390661346747036875, 2.68545256378962616170519853017, 3.55973890271065193943932726121, 3.62152119197582334243797673319, 4.13932672971655623186674626456, 4.23458362790248218519386092577, 4.88562457715919204178376484130, 5.11041422887724138233122285133, 5.85257914777428817189940596320, 5.85588904112827243672923891660, 6.42284192321717828423830798555, 6.52995576064869421005831147608, 7.23921495253468541444323295624, 7.28517638205992651476170625719, 7.88840432579027661822205047037, 7.994920884572981285922036248461
