L(s) = 1 | − 5-s + 7-s + 11-s + 13-s + 17-s − 19-s + 23-s + 25-s + 29-s + 31-s − 35-s + 37-s + 41-s + 43-s − 47-s + 49-s − 53-s − 55-s − 59-s − 61-s − 65-s + 67-s + 71-s − 73-s + 77-s − 79-s − 83-s + ⋯ |
L(s) = 1 | − 5-s + 7-s + 11-s + 13-s + 17-s − 19-s + 23-s + 25-s + 29-s + 31-s − 35-s + 37-s + 41-s + 43-s − 47-s + 49-s − 53-s − 55-s − 59-s − 61-s − 65-s + 67-s + 71-s − 73-s + 77-s − 79-s − 83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.177218963\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.177218963\) |
\(L(1)\) |
\(\approx\) |
\(1.253585913\) |
\(L(1)\) |
\(\approx\) |
\(1.253585913\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 \) |
good | 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 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| 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
−18.62916211780674271232877545409, −17.75224908734863499612528175610, −17.09109783475013815944598423638, −16.49703115039160750270802317857, −15.671446373660192132845357470781, −15.0915990990548757721853353764, −14.38898116542481719626536778749, −13.973874281316793229341714876651, −12.79335334985979740108599887169, −12.31471143838209683621554302546, −11.35604887662353531831423803324, −11.22367894923485098568128701866, −10.36449504864993709134611214287, −9.31661529826275114582653593119, −8.55313062708104404368647791547, −8.122014764267576813518216122027, −7.39326340811516523150798412808, −6.52116188496488693360609761736, −5.866137888116141320478958458151, −4.65930038018110762276488039629, −4.362751478096098167039390348275, −3.48458011163534663217794598260, −2.69264655938549929476686282087, −1.38636222308207391208670532206, −0.933874570424816596986164704999,
0.933874570424816596986164704999, 1.38636222308207391208670532206, 2.69264655938549929476686282087, 3.48458011163534663217794598260, 4.362751478096098167039390348275, 4.65930038018110762276488039629, 5.866137888116141320478958458151, 6.52116188496488693360609761736, 7.39326340811516523150798412808, 8.122014764267576813518216122027, 8.55313062708104404368647791547, 9.31661529826275114582653593119, 10.36449504864993709134611214287, 11.22367894923485098568128701866, 11.35604887662353531831423803324, 12.31471143838209683621554302546, 12.79335334985979740108599887169, 13.973874281316793229341714876651, 14.38898116542481719626536778749, 15.0915990990548757721853353764, 15.671446373660192132845357470781, 16.49703115039160750270802317857, 17.09109783475013815944598423638, 17.75224908734863499612528175610, 18.62916211780674271232877545409