| L(s) = 1 | + 5-s − 2·7-s − 2·13-s + 6·17-s + 10·19-s + 3·23-s − 6·29-s − 5·31-s − 2·35-s + 4·37-s + 12·41-s − 8·43-s − 12·47-s + 7·49-s + 6·53-s + 6·59-s + 7·61-s − 2·65-s − 2·67-s − 24·71-s − 32·73-s + 79-s − 15·83-s + 6·85-s + 24·89-s + 4·91-s + 10·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 0.554·13-s + 1.45·17-s + 2.29·19-s + 0.625·23-s − 1.11·29-s − 0.898·31-s − 0.338·35-s + 0.657·37-s + 1.87·41-s − 1.21·43-s − 1.75·47-s + 49-s + 0.824·53-s + 0.781·59-s + 0.896·61-s − 0.248·65-s − 0.244·67-s − 2.84·71-s − 3.74·73-s + 0.112·79-s − 1.64·83-s + 0.650·85-s + 2.54·89-s + 0.419·91-s + 1.02·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.323717615\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.323717615\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 12 T + 103 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 16 T + 159 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−9.627096946625867461752248566673, −9.229878683424296962663505641322, −8.942810603845708943826866275967, −8.540608620559694123648541416883, −7.79115672846270028165862531331, −7.42353419162335916711038543214, −7.34593049514981749829029524624, −7.02402798258473775936452474587, −6.15296500148472383129224226925, −5.86131873523810069205181729905, −5.64784628382244551014933536032, −5.15833923064090687449860109703, −4.70216987705528096228663152106, −4.11685894741917049189733146477, −3.31939411859317928661593182603, −3.28028393323370994930025626178, −2.78976220853864652860273553189, −1.96884560660324572121077180159, −1.34170110184150463463131333010, −0.63043147536621173449072760429,
0.63043147536621173449072760429, 1.34170110184150463463131333010, 1.96884560660324572121077180159, 2.78976220853864652860273553189, 3.28028393323370994930025626178, 3.31939411859317928661593182603, 4.11685894741917049189733146477, 4.70216987705528096228663152106, 5.15833923064090687449860109703, 5.64784628382244551014933536032, 5.86131873523810069205181729905, 6.15296500148472383129224226925, 7.02402798258473775936452474587, 7.34593049514981749829029524624, 7.42353419162335916711038543214, 7.79115672846270028165862531331, 8.540608620559694123648541416883, 8.942810603845708943826866275967, 9.229878683424296962663505641322, 9.627096946625867461752248566673
