| L(s) = 1 | + (−2 − 1.73i)7-s − 6.92i·13-s − 8·19-s + 5·25-s + 4·31-s − 10·37-s + 10.3i·43-s + (1.00 + 6.92i)49-s − 6.92i·61-s + 3.46i·67-s + 13.8i·73-s + 17.3i·79-s + (−11.9 + 13.8i)91-s − 13.8i·97-s − 20·103-s + ⋯ |
| L(s) = 1 | + (−0.755 − 0.654i)7-s − 1.92i·13-s − 1.83·19-s + 25-s + 0.718·31-s − 1.64·37-s + 1.58i·43-s + (0.142 + 0.989i)49-s − 0.887i·61-s + 0.423i·67-s + 1.62i·73-s + 1.94i·79-s + (−1.25 + 1.45i)91-s − 1.40i·97-s − 1.97·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
| good | 5 | \( 1 - 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 6.92iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 10.3iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 3.46iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 13.8iT - 73T^{2} \) |
| 79 | \( 1 - 17.3iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 13.8iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.199566610973997262487972070111, −7.15357294547050338895918910361, −6.58181770039281499252365128636, −5.84014921552629069172734818786, −4.98251416118862590997202031249, −4.11609856054723641509821858063, −3.25090034672791366186407817502, −2.56900346430565743076673182439, −1.09845867473557021305561096251, 0,
1.76137091916812079568454770161, 2.46113766620401014954185029841, 3.56036014806388508365427479140, 4.33925263852125342035115981391, 5.10565635794863763151346396162, 6.19195511296227158194415673406, 6.60849783931905347685963033106, 7.20262622107238898991785716907, 8.462364560816781876914209200498
