L(s) = 1 | + (1.30 − 1.30i)3-s + (−1 + i)5-s + (−1.30 − 1.30i)7-s − 0.394i·9-s + (3.30 + 3.30i)11-s − 4.60·13-s + 2.60i·15-s + (−3.60 + 2i)17-s − 6.60i·19-s − 3.39·21-s + (−0.697 − 0.697i)23-s + 3i·25-s + (3.39 + 3.39i)27-s + (5.60 − 5.60i)29-s + (0.697 − 0.697i)31-s + ⋯ |
L(s) = 1 | + (0.752 − 0.752i)3-s + (−0.447 + 0.447i)5-s + (−0.492 − 0.492i)7-s − 0.131i·9-s + (0.995 + 0.995i)11-s − 1.27·13-s + 0.672i·15-s + (−0.874 + 0.485i)17-s − 1.51i·19-s − 0.740·21-s + (−0.145 − 0.145i)23-s + 0.600i·25-s + (0.653 + 0.653i)27-s + (1.04 − 1.04i)29-s + (0.125 − 0.125i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.390i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 + 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.986939 - 0.200803i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.986939 - 0.200803i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + (3.60 - 2i)T \) |
good | 3 | \( 1 + (-1.30 + 1.30i)T - 3iT^{2} \) |
| 5 | \( 1 + (1 - i)T - 5iT^{2} \) |
| 7 | \( 1 + (1.30 + 1.30i)T + 7iT^{2} \) |
| 11 | \( 1 + (-3.30 - 3.30i)T + 11iT^{2} \) |
| 13 | \( 1 + 4.60T + 13T^{2} \) |
| 19 | \( 1 + 6.60iT - 19T^{2} \) |
| 23 | \( 1 + (0.697 + 0.697i)T + 23iT^{2} \) |
| 29 | \( 1 + (-5.60 + 5.60i)T - 29iT^{2} \) |
| 31 | \( 1 + (-0.697 + 0.697i)T - 31iT^{2} \) |
| 37 | \( 1 + (-3 + 3i)T - 37iT^{2} \) |
| 41 | \( 1 + (-1 - i)T + 41iT^{2} \) |
| 43 | \( 1 + 10.6iT - 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 - 9.21iT - 53T^{2} \) |
| 59 | \( 1 + 1.39iT - 59T^{2} \) |
| 61 | \( 1 + (-8.21 - 8.21i)T + 61iT^{2} \) |
| 67 | \( 1 + 5.21T + 67T^{2} \) |
| 71 | \( 1 + (7.90 - 7.90i)T - 71iT^{2} \) |
| 73 | \( 1 + (7 - 7i)T - 73iT^{2} \) |
| 79 | \( 1 + (-3.30 - 3.30i)T + 79iT^{2} \) |
| 83 | \( 1 + 3.81iT - 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 + (0.394 - 0.394i)T - 97iT^{2} \) |
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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
−14.67343033648117018520552935583, −13.61842885634957458770977915766, −12.69960767388165359911254253095, −11.54763284687966273074464065341, −10.09053672591654157902121285853, −8.866188791384350709477392804753, −7.34951524362021008177013726151, −6.85860801102215847954731666093, −4.35907414438787594393946622577, −2.49084501574773422343500138397,
3.15701690411250357516773282093, 4.53078632961059137419158729594, 6.37887343795492470595592728260, 8.216623488955160057307270900011, 9.120483433572484640649899245151, 10.02010890216736633778116569053, 11.68509971788769434360472949046, 12.54405286762386589355211514815, 14.12625175538854920509090204593, 14.77562005464085257895993916105