L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−1.59 − 2.75i)5-s + 0.999·8-s + 3.18·10-s + (1.59 − 2.75i)11-s + (2.85 + 4.93i)13-s + (−0.5 + 0.866i)16-s − 1.52·17-s + 1.28·19-s + (−1.59 + 2.75i)20-s + (1.59 + 2.75i)22-s + (1.11 + 1.93i)23-s + (−2.56 + 4.43i)25-s − 5.70·26-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.711 − 1.23i)5-s + 0.353·8-s + 1.00·10-s + (0.479 − 0.830i)11-s + (0.790 + 1.36i)13-s + (−0.125 + 0.216i)16-s − 0.369·17-s + 0.294·19-s + (−0.355 + 0.616i)20-s + (0.339 + 0.587i)22-s + (0.233 + 0.404i)23-s + (−0.512 + 0.887i)25-s − 1.11·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.290194708\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.290194708\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (1.59 + 2.75i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.59 + 2.75i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.85 - 4.93i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 1.52T + 17T^{2} \) |
| 19 | \( 1 - 1.28T + 19T^{2} \) |
| 23 | \( 1 + (-1.11 - 1.93i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.54 + 6.13i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.71 - 8.15i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + (-2.80 - 4.85i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.41 + 5.91i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2.91 - 5.04i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 2.05T + 53T^{2} \) |
| 59 | \( 1 + (0.562 + 0.974i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.56 - 2.70i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.48 + 9.49i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 8.69T + 71T^{2} \) |
| 73 | \( 1 - 4.96T + 73T^{2} \) |
| 79 | \( 1 + (-2.06 + 3.58i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.03 + 6.98i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 0.225T + 89T^{2} \) |
| 97 | \( 1 + (-7.42 + 12.8i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.844234788077332678070939187798, −8.231538448482569357457288435897, −7.40843896144145991125423341826, −6.48102687314606775512422788392, −5.91121712787773233804602611568, −4.74587265709566325901222070831, −4.34218231617599454944520556693, −3.32206102315913919660356818385, −1.60549578760723316103787683463, −0.69472788882364192771843382213,
0.887267577064356942672359299640, 2.34493898517393204812768441220, 3.14138220163184514049850177015, 3.83226101871161574634998952791, 4.74045740902625792604705406984, 5.95810397542031422095050929134, 6.79737107427782640364970174725, 7.46757129269369439629814537184, 8.120313196972473474280961724710, 8.902268410550725599366711609350