| L(s) = 1 | − 2·5-s − 4·7-s − 2·9-s + 8·19-s + 3·25-s + 8·35-s + 4·37-s + 20·43-s + 4·45-s − 2·49-s − 12·53-s + 8·63-s − 16·79-s − 5·81-s − 12·83-s − 12·89-s − 16·95-s + 4·97-s + 12·107-s − 12·113-s − 11·121-s − 4·125-s + 127-s + 131-s − 32·133-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.51·7-s − 2/3·9-s + 1.83·19-s + 3/5·25-s + 1.35·35-s + 0.657·37-s + 3.04·43-s + 0.596·45-s − 2/7·49-s − 1.64·53-s + 1.00·63-s − 1.80·79-s − 5/9·81-s − 1.31·83-s − 1.27·89-s − 1.64·95-s + 0.406·97-s + 1.16·107-s − 1.12·113-s − 121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s − 2.77·133-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.740293324354028093320034492248, −8.597075432672291247955228431981, −7.71273110823499542846308181544, −7.48339952967373098458922175110, −7.12025897518162428207793885558, −6.27087624192875571051851265258, −6.09088770920212572077792597538, −5.46752609716119299096669229449, −4.83331995302090139893539895632, −4.12250433368686324236236368171, −3.61120105048083329633068803983, −2.90312545354050643736888796120, −2.76929890617261215013507568311, −1.17768193714929585417069498904, 0,
1.17768193714929585417069498904, 2.76929890617261215013507568311, 2.90312545354050643736888796120, 3.61120105048083329633068803983, 4.12250433368686324236236368171, 4.83331995302090139893539895632, 5.46752609716119299096669229449, 6.09088770920212572077792597538, 6.27087624192875571051851265258, 7.12025897518162428207793885558, 7.48339952967373098458922175110, 7.71273110823499542846308181544, 8.597075432672291247955228431981, 8.740293324354028093320034492248
