| L(s) = 1 | + 2·5-s + 9-s − 4·11-s + 8·23-s + 3·25-s + 16·31-s − 4·37-s + 2·45-s − 8·47-s + 2·49-s + 12·53-s − 8·55-s + 8·59-s + 81-s + 4·89-s − 12·97-s − 4·99-s − 8·103-s − 12·113-s + 16·115-s + 5·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1/3·9-s − 1.20·11-s + 1.66·23-s + 3/5·25-s + 2.87·31-s − 0.657·37-s + 0.298·45-s − 1.16·47-s + 2/7·49-s + 1.64·53-s − 1.07·55-s + 1.04·59-s + 1/9·81-s + 0.423·89-s − 1.21·97-s − 0.402·99-s − 0.788·103-s − 1.12·113-s + 1.49·115-s + 5/11·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.732592328\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.732592328\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−7.909284291233105168622459188957, −7.30940145413148995099396888146, −6.89944682733768645934258523854, −6.62594133075902688720658920775, −6.14686405617528814661647415817, −5.55602485863135052462085349924, −5.22362898947478958246795042556, −4.87548114255193463330239361979, −4.40838433666248209459559516555, −3.77582647930111331905303414263, −2.97432235642538657105544786476, −2.75769459560026897324597618232, −2.21924034969923772893700992224, −1.39289629140132360968414942660, −0.73239427623541784610581367811,
0.73239427623541784610581367811, 1.39289629140132360968414942660, 2.21924034969923772893700992224, 2.75769459560026897324597618232, 2.97432235642538657105544786476, 3.77582647930111331905303414263, 4.40838433666248209459559516555, 4.87548114255193463330239361979, 5.22362898947478958246795042556, 5.55602485863135052462085349924, 6.14686405617528814661647415817, 6.62594133075902688720658920775, 6.89944682733768645934258523854, 7.30940145413148995099396888146, 7.909284291233105168622459188957
