L(s) = 1 | − 2-s + (−1.5 + 0.866i)3-s + 4-s − 1.73i·5-s + (1.5 − 0.866i)6-s + (−2.5 − 0.866i)7-s − 8-s + (1.5 − 2.59i)9-s + 1.73i·10-s + (−1.5 + 0.866i)12-s + (1 + 3.46i)13-s + (2.5 + 0.866i)14-s + (1.49 + 2.59i)15-s + 16-s − 3·17-s + (−1.5 + 2.59i)18-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.866 + 0.499i)3-s + 0.5·4-s − 0.774i·5-s + (0.612 − 0.353i)6-s + (−0.944 − 0.327i)7-s − 0.353·8-s + (0.5 − 0.866i)9-s + 0.547i·10-s + (−0.433 + 0.249i)12-s + (0.277 + 0.960i)13-s + (0.668 + 0.231i)14-s + (0.387 + 0.670i)15-s + 0.250·16-s − 0.727·17-s + (−0.353 + 0.612i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.453 - 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.176029 + 0.287228i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.176029 + 0.287228i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (1.5 - 0.866i)T \) |
| 7 | \( 1 + (2.5 + 0.866i)T \) |
| 13 | \( 1 + (-1 - 3.46i)T \) |
good | 5 | \( 1 + 1.73iT - 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 3.46iT - 23T^{2} \) |
| 29 | \( 1 - 6.92iT - 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 1.73iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 11T + 43T^{2} \) |
| 47 | \( 1 - 12.1iT - 47T^{2} \) |
| 53 | \( 1 - 3.46iT - 53T^{2} \) |
| 59 | \( 1 - 10.3iT - 59T^{2} \) |
| 61 | \( 1 + 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 13.8iT - 67T^{2} \) |
| 71 | \( 1 + 9T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 3.46iT - 83T^{2} \) |
| 89 | \( 1 + 3.46iT - 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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
−10.98744514629988858298932280200, −10.16811071041008149835523994473, −9.184848695926766363294939809326, −8.967462412349409188519111699868, −7.33273285142611350748768584801, −6.62202373195864845150671672461, −5.63466075184820213684113710937, −4.53724676716970795690871960480, −3.41251745273826628381294344249, −1.34458609380992478097970620333,
0.28739572070691110036812848507, 2.21039341179798672450867367705, 3.42246719177726680751014629698, 5.21843143127830986396278176720, 6.26677459256349196366371373804, 6.77010883631986819751188178852, 7.69830171852438734904832294612, 8.726364576074737566685452204941, 9.902753679929592513909221530771, 10.47940537339231293610537384530