L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + i·7-s + 8-s + 9-s − i·11-s − 12-s + i·13-s + i·14-s + 16-s + 17-s + 18-s + i·19-s + ⋯ |
L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + i·7-s + 8-s + 9-s − i·11-s − 12-s + i·13-s + i·14-s + 16-s + 17-s + 18-s + i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.526411005 + 0.2803797537i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.526411005 + 0.2803797537i\) |
\(L(1)\) |
\(\approx\) |
\(1.445877376 + 0.1421279466i\) |
\(L(1)\) |
\(\approx\) |
\(1.445877376 + 0.1421279466i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + iT \) |
| 23 | \( 1 + T \) |
| 31 | \( 1 \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 + iT \) |
| 53 | \( 1 \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + iT \) |
| 73 | \( 1 \) |
| 79 | \( 1 - iT \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 - iT \) |
| 97 | \( 1 - T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−28.30322975755651199153291704508, −27.48438060800441617218160639788, −26.02897984295937491936307082715, −25.00185387492234196054944632173, −23.801737446415263314627299028331, −23.1949714272981043099428408158, −22.530312014516157291647615828028, −21.43184146972020823105848749063, −20.45008214323424230750962793573, −19.48777878421408130500621982103, −17.77959849512311369916281998332, −17.06802709002563636570938338282, −15.94130584297147392423052031316, −15.05721319979724988861102300850, −13.700537149836299065565886274440, −12.77029500395329630977375903801, −11.88821601792367444428125804610, −10.715215028499248906273346095002, −10.01732818063003308736840426979, −7.558547676080904308594241522415, −6.886327969630382692155581992860, −5.50150836053284073336478971035, −4.65383926034041601073026630363, −3.38521264351522945136246879010, −1.40107555267379478213052944477,
1.76058535081482654558672248219, 3.43233880572970114448586214940, 4.82720010917748641348470847112, 5.84009431590266780401976831588, 6.5447175830808968273790855940, 8.12238340655343187603806381914, 9.880439347391169883791694346751, 11.174147059515731578083194644448, 11.91275923756736253828454527806, 12.703334850359798019100488694926, 14.01457590587866034083915492254, 15.075517524403959638932259213152, 16.327365445046418472746817567576, 16.69518655322467477809083356036, 18.482464000858932682744469460362, 19.11597165039085724449461550869, 20.93904169332364768427162130737, 21.51372291747184055668055407656, 22.37731131504355664204917169322, 23.25878321034084280548333274843, 24.25679076388511478548809201219, 24.85742853991052127018854336503, 26.23466724132784587978975245008, 27.608611311885133532323631759423, 28.575415900258540769070171878842