| L(s) = 1 | − 4·3-s + 4·5-s + 6·9-s − 16·15-s + 11·25-s + 4·27-s + 8·31-s + 12·37-s − 4·41-s − 16·43-s + 24·45-s + 49-s + 20·53-s + 24·67-s − 44·75-s − 16·79-s − 37·81-s − 12·83-s + 20·89-s − 32·93-s + 24·107-s − 48·111-s − 22·121-s + 16·123-s + 24·125-s + 127-s + 64·129-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 1.78·5-s + 2·9-s − 4.13·15-s + 11/5·25-s + 0.769·27-s + 1.43·31-s + 1.97·37-s − 0.624·41-s − 2.43·43-s + 3.57·45-s + 1/7·49-s + 2.74·53-s + 2.93·67-s − 5.08·75-s − 1.80·79-s − 4.11·81-s − 1.31·83-s + 2.11·89-s − 3.31·93-s + 2.32·107-s − 4.55·111-s − 2·121-s + 1.44·123-s + 2.14·125-s + 0.0887·127-s + 5.63·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 627200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 627200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.136826477\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.136826477\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.589748372800265236248597894315, −7.965971686397333220186526353962, −7.00881059716814921000381829301, −6.86591116390976664813936408049, −6.25148198523199051609095248236, −6.19535765189485358396007496237, −5.55399016745426097333734158356, −5.43110360120582172209460233834, −4.80018849900442151332658226553, −4.61889361197427675528739680931, −3.62433266137155427382097354955, −2.73137510590455017802711867793, −2.29482126376589182159266782001, −1.29582036582827735540991947956, −0.68915328699946070952810996679,
0.68915328699946070952810996679, 1.29582036582827735540991947956, 2.29482126376589182159266782001, 2.73137510590455017802711867793, 3.62433266137155427382097354955, 4.61889361197427675528739680931, 4.80018849900442151332658226553, 5.43110360120582172209460233834, 5.55399016745426097333734158356, 6.19535765189485358396007496237, 6.25148198523199051609095248236, 6.86591116390976664813936408049, 7.00881059716814921000381829301, 7.965971686397333220186526353962, 8.589748372800265236248597894315
