| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s − 11-s − 13-s + 15-s + 17-s + 19-s + 21-s + 23-s + 25-s + 27-s − 29-s + 31-s − 33-s + 35-s − 37-s − 39-s − 41-s + 43-s + 45-s − 47-s + 49-s + 51-s + 53-s − 55-s + ⋯ |
| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s − 11-s − 13-s + 15-s + 17-s + 19-s + 21-s + 23-s + 25-s + 27-s − 29-s + 31-s − 33-s + 35-s − 37-s − 39-s − 41-s + 43-s + 45-s − 47-s + 49-s + 51-s + 53-s − 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 356 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 356 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(4.008703883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.008703883\) |
| \(L(1)\) |
\(\approx\) |
\(1.998048931\) |
| \(L(1)\) |
\(\approx\) |
\(1.998048931\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 89 | \( 1 \) |
| good | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| 97 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−24.54511626279404589407055550274, −24.11614858040806048459448344140, −22.70312795124893878093629019008, −21.55510505359097998575344680854, −20.95739453842689344259634930333, −20.47419034543283728978604234746, −19.17541101628939639394319050766, −18.38592102150349512157180482245, −17.58460284605139119925376956450, −16.589346534493182262117330941449, −15.29999012014942989352861397416, −14.57987026089759079567372495166, −13.8366438257837443287992753916, −13.05569432276474049292097652817, −11.96464959247643322801255432238, −10.52537025533763565778346299510, −9.84998455095688825031696259499, −8.89090648833392100651767815820, −7.834842473214199709705392030933, −7.14535978999608023815317379685, −5.44911508873303616975698110805, −4.81980556918107507420946662605, −3.17276477077054555167466274689, −2.28443402015167398577598064602, −1.24490721113590129679735624215,
1.24490721113590129679735624215, 2.28443402015167398577598064602, 3.17276477077054555167466274689, 4.81980556918107507420946662605, 5.44911508873303616975698110805, 7.14535978999608023815317379685, 7.834842473214199709705392030933, 8.89090648833392100651767815820, 9.84998455095688825031696259499, 10.52537025533763565778346299510, 11.96464959247643322801255432238, 13.05569432276474049292097652817, 13.8366438257837443287992753916, 14.57987026089759079567372495166, 15.29999012014942989352861397416, 16.589346534493182262117330941449, 17.58460284605139119925376956450, 18.38592102150349512157180482245, 19.17541101628939639394319050766, 20.47419034543283728978604234746, 20.95739453842689344259634930333, 21.55510505359097998575344680854, 22.70312795124893878093629019008, 24.11614858040806048459448344140, 24.54511626279404589407055550274
