L(s) = 1 | − 2-s + 4-s − 1.61·5-s − 8-s + 9-s + 1.61·10-s − 0.618·13-s + 16-s + 0.618·17-s − 18-s − 1.61·20-s + 1.61·25-s + 0.618·26-s − 0.618·29-s − 32-s − 0.618·34-s + 36-s + 1.61·37-s + 1.61·40-s + 1.61·41-s − 1.61·45-s + 49-s − 1.61·50-s − 0.618·52-s + 1.61·53-s + 0.618·58-s + 0.618·61-s + ⋯ |
L(s) = 1 | − 2-s + 4-s − 1.61·5-s − 8-s + 9-s + 1.61·10-s − 0.618·13-s + 16-s + 0.618·17-s − 18-s − 1.61·20-s + 1.61·25-s + 0.618·26-s − 0.618·29-s − 32-s − 0.618·34-s + 36-s + 1.61·37-s + 1.61·40-s + 1.61·41-s − 1.61·45-s + 49-s − 1.61·50-s − 0.618·52-s + 1.61·53-s + 0.618·58-s + 0.618·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5501939877\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5501939877\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 - T^{2} \) |
| 5 | \( 1 + 1.61T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 0.618T + T^{2} \) |
| 17 | \( 1 - 0.618T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 0.618T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 1.61T + T^{2} \) |
| 41 | \( 1 - 1.61T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 1.61T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 0.618T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.61T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.61T + T^{2} \) |
| 97 | \( 1 + 0.618T + T^{2} \) |
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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
−9.686355119728543959019669599620, −8.895717640035701762689097646290, −7.928169030090484556286769673947, −7.50724612509392176619659619870, −6.97024317239927086920734796609, −5.76541842432722038985123047858, −4.44979234967943188284752610360, −3.68880532658430484676578010246, −2.50894823879399534344941514697, −0.927496171838661277290974154215,
0.927496171838661277290974154215, 2.50894823879399534344941514697, 3.68880532658430484676578010246, 4.44979234967943188284752610360, 5.76541842432722038985123047858, 6.97024317239927086920734796609, 7.50724612509392176619659619870, 7.928169030090484556286769673947, 8.895717640035701762689097646290, 9.686355119728543959019669599620