L(s) = 1 | − 0.378i·7-s + (−2.46 − 2.46i)13-s + (−2.44 − 2.44i)19-s + 5i·25-s − 10.1·31-s + (1.53 − 1.53i)37-s + (−7.34 + 7.34i)43-s + 6.85·49-s + (−10.4 − 10.4i)61-s + (−11.3 − 11.3i)67-s + 13.8i·73-s − 9.41·79-s + (−0.933 + 0.933i)91-s + 13.8·97-s − 11.6i·103-s + ⋯ |
L(s) = 1 | − 0.143i·7-s + (−0.683 − 0.683i)13-s + (−0.561 − 0.561i)19-s + i·25-s − 1.82·31-s + (0.252 − 0.252i)37-s + (−1.12 + 1.12i)43-s + 0.979·49-s + (−1.33 − 1.33i)61-s + (−1.38 − 1.38i)67-s + 1.62i·73-s − 1.05·79-s + (−0.0978 + 0.0978i)91-s + 1.40·97-s − 1.15i·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2261058370\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2261058370\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 5iT^{2} \) |
| 7 | \( 1 + 0.378iT - 7T^{2} \) |
| 11 | \( 1 - 11iT^{2} \) |
| 13 | \( 1 + (2.46 + 2.46i)T + 13iT^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + (2.44 + 2.44i)T + 19iT^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 29iT^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 + (-1.53 + 1.53i)T - 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (7.34 - 7.34i)T - 43iT^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53iT^{2} \) |
| 59 | \( 1 - 59iT^{2} \) |
| 61 | \( 1 + (10.4 + 10.4i)T + 61iT^{2} \) |
| 67 | \( 1 + (11.3 + 11.3i)T + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 13.8iT - 73T^{2} \) |
| 79 | \( 1 + 9.41T + 79T^{2} \) |
| 83 | \( 1 + 83iT^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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
−8.718372190383889712243579614033, −7.71739118951589261216707981111, −7.25037574690372941158562362266, −6.29543954265114735027091037029, −5.41585950658977920837317491934, −4.70728173270438452221979721595, −3.66243766781013881886716977819, −2.77142668409070737687800926766, −1.62177224685869915673908173932, −0.07244231811309053405953560284,
1.67061549212658918453612669832, 2.58787161712220285390004411847, 3.76989677159229344038473931681, 4.53317229824779447960379692135, 5.47037131951553549638149302104, 6.26492720268980908782343801998, 7.10991523209489605594719789227, 7.76745872221598460646807678796, 8.764147730002048046160991426490, 9.197843585327449698378112847431