L(s) = 1 | + (−0.446 + 1.34i)2-s + (1.32 + 1.11i)3-s + (−1.60 − 1.19i)4-s − 0.803i·5-s + (−2.08 + 1.28i)6-s + (2.32 − 1.61i)8-s + (0.523 + 2.95i)9-s + (1.07 + 0.358i)10-s + 2.34·11-s + (−0.790 − 3.37i)12-s + 5.26·13-s + (0.893 − 1.06i)15-s + (1.12 + 3.83i)16-s − 1.18i·17-s + (−4.19 − 0.616i)18-s + 7.12i·19-s + ⋯ |
L(s) = 1 | + (−0.315 + 0.948i)2-s + (0.766 + 0.642i)3-s + (−0.800 − 0.599i)4-s − 0.359i·5-s + (−0.851 + 0.524i)6-s + (0.821 − 0.569i)8-s + (0.174 + 0.984i)9-s + (0.340 + 0.113i)10-s + 0.707·11-s + (−0.228 − 0.973i)12-s + 1.46·13-s + (0.230 − 0.275i)15-s + (0.281 + 0.959i)16-s − 0.287i·17-s + (−0.989 − 0.145i)18-s + 1.63i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.228 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.228 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.994415 + 1.25452i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.994415 + 1.25452i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.446 - 1.34i)T \) |
| 3 | \( 1 + (-1.32 - 1.11i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 0.803iT - 5T^{2} \) |
| 11 | \( 1 - 2.34T + 11T^{2} \) |
| 13 | \( 1 - 5.26T + 13T^{2} \) |
| 17 | \( 1 + 1.18iT - 17T^{2} \) |
| 19 | \( 1 - 7.12iT - 19T^{2} \) |
| 23 | \( 1 + 7.88T + 23T^{2} \) |
| 29 | \( 1 + 4.23iT - 29T^{2} \) |
| 31 | \( 1 - 4.89iT - 31T^{2} \) |
| 37 | \( 1 - 1.04T + 37T^{2} \) |
| 41 | \( 1 - 7.16iT - 41T^{2} \) |
| 43 | \( 1 + 7.94iT - 43T^{2} \) |
| 47 | \( 1 - 6.09T + 47T^{2} \) |
| 53 | \( 1 + 8.72iT - 53T^{2} \) |
| 59 | \( 1 - 0.662T + 59T^{2} \) |
| 61 | \( 1 - 0.958T + 61T^{2} \) |
| 67 | \( 1 - 8.42iT - 67T^{2} \) |
| 71 | \( 1 + 9.67T + 71T^{2} \) |
| 73 | \( 1 + 1.41T + 73T^{2} \) |
| 79 | \( 1 + 6.92iT - 79T^{2} \) |
| 83 | \( 1 - 5.18T + 83T^{2} \) |
| 89 | \( 1 + 16.3iT - 89T^{2} \) |
| 97 | \( 1 + 4.37T + 97T^{2} \) |
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
−10.45882286970038069152981732929, −9.932243128545215756963541552274, −8.898624842676191253884922670683, −8.446845652252436242243996956545, −7.66005840942626990231081806318, −6.38863957404294824357025738532, −5.56739755711247130062804550272, −4.30709876410858304079644065118, −3.63821863455214341698918663282, −1.55640685894692912115993054056,
1.11096850314094963335574941772, 2.36855282045604040986508131493, 3.44330661833155655373824996284, 4.28827662067451203866644383929, 6.05917206991751946399346190898, 7.04126874199770673863524077812, 8.050985465657887098046919086505, 8.856865270675677798309905201585, 9.372610626618935291576458378450, 10.56427461326749210167519925962