| L(s) = 1 | − 0.792·3-s + (−0.792 + 2.52i)7-s − 2.37·9-s + 0.792i·11-s + 5.37i·13-s + 3.37i·17-s + 3.46·19-s + (0.627 − 2i)21-s − 1.87i·23-s + 4.25·27-s − 5.37·29-s + 8.51·31-s − 0.627i·33-s + 0.744·37-s − 4.25i·39-s + ⋯ |
| L(s) = 1 | − 0.457·3-s + (−0.299 + 0.954i)7-s − 0.790·9-s + 0.238i·11-s + 1.49i·13-s + 0.817i·17-s + 0.794·19-s + (0.136 − 0.436i)21-s − 0.391i·23-s + 0.819·27-s − 0.997·29-s + 1.52·31-s − 0.109i·33-s + 0.122·37-s − 0.681i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.217i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 + 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4175427333\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4175427333\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.792 - 2.52i)T \) |
| good | 3 | \( 1 + 0.792T + 3T^{2} \) |
| 11 | \( 1 - 0.792iT - 11T^{2} \) |
| 13 | \( 1 - 5.37iT - 13T^{2} \) |
| 17 | \( 1 - 3.37iT - 17T^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 23 | \( 1 + 1.87iT - 23T^{2} \) |
| 29 | \( 1 + 5.37T + 29T^{2} \) |
| 31 | \( 1 - 8.51T + 31T^{2} \) |
| 37 | \( 1 - 0.744T + 37T^{2} \) |
| 41 | \( 1 + 2.74iT - 41T^{2} \) |
| 43 | \( 1 + 3.46iT - 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 + 6.63T + 59T^{2} \) |
| 61 | \( 1 + 0.744iT - 61T^{2} \) |
| 67 | \( 1 + 6.63iT - 67T^{2} \) |
| 71 | \( 1 - 6.63iT - 71T^{2} \) |
| 73 | \( 1 - 2.74iT - 73T^{2} \) |
| 79 | \( 1 + 14.0iT - 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 - 17.4iT - 89T^{2} \) |
| 97 | \( 1 + 2.11iT - 97T^{2} \) |
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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
−9.246820685363879021961045340630, −8.512602448219325974204352552585, −7.79892548895601724197447732496, −6.55727845313337248179451944063, −6.31988505849604139424157448093, −5.37115635960049186147429476553, −4.67861741300313068782223262669, −3.58778909273503332959460357386, −2.60963509773704952971045651218, −1.63440004148500708897936615077,
0.15567829443344706866585458359, 1.14538804912830646473759777609, 2.92719002272856136711579585722, 3.31453369834933869677226052714, 4.61455816223052971347908095713, 5.32120767684463593591394601887, 6.06619796236144133213621445224, 6.81331600538590722761882877823, 7.81302026346778224881572687080, 8.098113230914259639240293667938
