| L(s) = 1 | − 2·3-s + 2·5-s − 4·7-s + 9-s − 4·15-s − 8·17-s + 8·21-s − 25-s + 4·27-s − 8·35-s − 16·37-s − 8·41-s + 8·43-s + 2·45-s − 8·47-s + 9·49-s + 16·51-s − 4·63-s − 8·67-s + 2·75-s + 16·79-s − 11·81-s + 12·83-s − 16·85-s + 8·89-s − 16·101-s + 16·105-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s − 1.51·7-s + 1/3·9-s − 1.03·15-s − 1.94·17-s + 1.74·21-s − 1/5·25-s + 0.769·27-s − 1.35·35-s − 2.63·37-s − 1.24·41-s + 1.21·43-s + 0.298·45-s − 1.16·47-s + 9/7·49-s + 2.24·51-s − 0.503·63-s − 0.977·67-s + 0.230·75-s + 1.80·79-s − 1.22·81-s + 1.31·83-s − 1.73·85-s + 0.847·89-s − 1.59·101-s + 1.56·105-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3636507417\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3636507417\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−8.456010477067388384235063839284, −7.83128189768373739713825649327, −7.07444990030028251967115790143, −6.75258247045954677992796922996, −6.49604506438419960021163135817, −6.16091699780800019349013994634, −5.68113243776708285168897607321, −5.09143981565822196282912576073, −4.90019236484334859195739888494, −4.04868782580466863561704614573, −3.56786087093513565311335009336, −2.90110894756278118503116162634, −2.25997635543693402518217423275, −1.60587094068944614124427073336, −0.30578820314114231472897285365,
0.30578820314114231472897285365, 1.60587094068944614124427073336, 2.25997635543693402518217423275, 2.90110894756278118503116162634, 3.56786087093513565311335009336, 4.04868782580466863561704614573, 4.90019236484334859195739888494, 5.09143981565822196282912576073, 5.68113243776708285168897607321, 6.16091699780800019349013994634, 6.49604506438419960021163135817, 6.75258247045954677992796922996, 7.07444990030028251967115790143, 7.83128189768373739713825649327, 8.456010477067388384235063839284
