L(s) = 1 | + (0.796 + 1.16i)2-s − 3-s + (−0.731 + 1.86i)4-s + (−0.796 − 1.16i)6-s − 4.72i·7-s + (−2.75 + 0.627i)8-s + 9-s − 3.93i·11-s + (0.731 − 1.86i)12-s + 3.46·13-s + (5.51 − 3.76i)14-s + (−2.93 − 2.72i)16-s − 3.51i·17-s + (0.796 + 1.16i)18-s + 5.44i·19-s + ⋯ |
L(s) = 1 | + (0.563 + 0.826i)2-s − 0.577·3-s + (−0.365 + 0.930i)4-s + (−0.325 − 0.477i)6-s − 1.78i·7-s + (−0.975 + 0.221i)8-s + 0.333·9-s − 1.18i·11-s + (0.211 − 0.537i)12-s + 0.961·13-s + (1.47 − 1.00i)14-s + (−0.732 − 0.680i)16-s − 0.852i·17-s + (0.187 + 0.275i)18-s + 1.24i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.237i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 + 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.40395 - 0.169161i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.40395 - 0.169161i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.796 - 1.16i)T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 4.72iT - 7T^{2} \) |
| 11 | \( 1 + 3.93iT - 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 + 3.51iT - 17T^{2} \) |
| 19 | \( 1 - 5.44iT - 19T^{2} \) |
| 23 | \( 1 + 7.11iT - 23T^{2} \) |
| 29 | \( 1 - 3.66iT - 29T^{2} \) |
| 31 | \( 1 - 5.23T + 31T^{2} \) |
| 37 | \( 1 + 0.414T + 37T^{2} \) |
| 41 | \( 1 - 3.00T + 41T^{2} \) |
| 43 | \( 1 + 5.34T + 43T^{2} \) |
| 47 | \( 1 - 0.925iT - 47T^{2} \) |
| 53 | \( 1 - 0.233T + 53T^{2} \) |
| 59 | \( 1 + 14.3iT - 59T^{2} \) |
| 61 | \( 1 + 0.118iT - 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 - 2.19T + 71T^{2} \) |
| 73 | \( 1 + 0.563iT - 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 + 8.88T + 89T^{2} \) |
| 97 | \( 1 + 7.27iT - 97T^{2} \) |
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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
−10.78940821858087796767321753899, −9.871738201482319245387567324551, −8.504991852577204139014190162819, −7.86412382185461724500452979005, −6.76950977940800217771715952274, −6.28043033993528495516403266436, −5.12232335622226945838883861374, −4.11684940789948159194716655586, −3.34857366994614802628597461169, −0.75378782381264925568275376412,
1.65741592240601726137252813628, 2.73613347639555669539913242745, 4.12944620590399718608812948411, 5.18888928974745412117083229895, 5.85020378249386504780783326119, 6.73463306743801881842293650224, 8.322106292135072679998346309584, 9.247816033024984076394132220388, 9.874909655210862644928444077086, 10.98562098588897618690806347033