L(s) = 1 | + 3-s + 2·5-s + 9-s − 11-s − 13-s + 2·15-s + 2·17-s − 4·19-s − 8·23-s − 25-s + 27-s − 6·29-s + 4·31-s − 33-s − 6·37-s − 39-s − 2·41-s − 8·43-s + 2·45-s − 12·47-s − 7·49-s + 2·51-s + 14·53-s − 2·55-s − 4·57-s − 12·59-s + 2·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s − 0.301·11-s − 0.277·13-s + 0.516·15-s + 0.485·17-s − 0.917·19-s − 1.66·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.174·33-s − 0.986·37-s − 0.160·39-s − 0.312·41-s − 1.21·43-s + 0.298·45-s − 1.75·47-s − 49-s + 0.280·51-s + 1.92·53-s − 0.269·55-s − 0.529·57-s − 1.56·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.916183736147810382535573288840, −6.82568833118493832246682107957, −6.25197304711568605822227654909, −5.52705307715950068574439605922, −4.78872014659826137638267136904, −3.88309824552053572971723317270, −3.13150081304305909354021709729, −2.09252086470755029926420894000, −1.70735315121739404640448403280, 0,
1.70735315121739404640448403280, 2.09252086470755029926420894000, 3.13150081304305909354021709729, 3.88309824552053572971723317270, 4.78872014659826137638267136904, 5.52705307715950068574439605922, 6.25197304711568605822227654909, 6.82568833118493832246682107957, 7.916183736147810382535573288840