L(s) = 1 | + 1.56i·2-s + (0.707 − 0.707i)3-s − 0.438·4-s + (−0.397 + 0.397i)5-s + (1.10 + 1.10i)6-s + 2.43i·8-s − 1.00i·9-s + (−0.620 − 0.620i)10-s + (1.81 + 1.81i)11-s + (−0.310 + 0.310i)12-s − 4.56·13-s + 0.561i·15-s − 4.68·16-s + 1.56·18-s + 7.68i·19-s + (0.174 − 0.174i)20-s + ⋯ |
L(s) = 1 | + 1.10i·2-s + (0.408 − 0.408i)3-s − 0.219·4-s + (−0.177 + 0.177i)5-s + (0.450 + 0.450i)6-s + 0.862i·8-s − 0.333i·9-s + (−0.196 − 0.196i)10-s + (0.546 + 0.546i)11-s + (−0.0894 + 0.0894i)12-s − 1.26·13-s + 0.144i·15-s − 1.17·16-s + 0.368·18-s + 1.76i·19-s + (0.0389 − 0.0389i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.591 - 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.591 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.784230 + 1.54900i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.784230 + 1.54900i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 17 | \( 1 \) |
good | 2 | \( 1 - 1.56iT - 2T^{2} \) |
| 5 | \( 1 + (0.397 - 0.397i)T - 5iT^{2} \) |
| 7 | \( 1 + 7iT^{2} \) |
| 11 | \( 1 + (-1.81 - 1.81i)T + 11iT^{2} \) |
| 13 | \( 1 + 4.56T + 13T^{2} \) |
| 19 | \( 1 - 7.68iT - 19T^{2} \) |
| 23 | \( 1 + (-4.63 - 4.63i)T + 23iT^{2} \) |
| 29 | \( 1 + (-5.83 + 5.83i)T - 29iT^{2} \) |
| 31 | \( 1 + (-3.62 + 3.62i)T - 31iT^{2} \) |
| 37 | \( 1 + (2.20 - 2.20i)T - 37iT^{2} \) |
| 41 | \( 1 + (-0.397 - 0.397i)T + 41iT^{2} \) |
| 43 | \( 1 - 7.68iT - 43T^{2} \) |
| 47 | \( 1 - 2.87T + 47T^{2} \) |
| 53 | \( 1 + 4.24iT - 53T^{2} \) |
| 59 | \( 1 - 1.12iT - 59T^{2} \) |
| 61 | \( 1 + (-0.620 - 0.620i)T + 61iT^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + (7.24 - 7.24i)T - 71iT^{2} \) |
| 73 | \( 1 + (-3.00 + 3.00i)T - 73iT^{2} \) |
| 79 | \( 1 + (10.8 + 10.8i)T + 79iT^{2} \) |
| 83 | \( 1 + 9.12iT - 83T^{2} \) |
| 89 | \( 1 + 7.12T + 89T^{2} \) |
| 97 | \( 1 + (7.86 - 7.86i)T - 97iT^{2} \) |
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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
−10.16287536711003636413729208477, −9.496190225524007565746118287404, −8.424002603756086734336087305694, −7.68520165489123436471615856641, −7.17089586301491720923208184193, −6.34493682245350177060364436048, −5.42611046196853657329843868792, −4.35652771637415780562675744485, −2.99420767330911308405786532413, −1.76431050281635799394570200718,
0.806936643996667859185094636260, 2.47901014337561795061079821897, 3.04586648960022745387223518739, 4.31395888440212386479391077057, 4.97354320747223459605490161897, 6.60527588486031017573620795418, 7.22119716927976604112161777176, 8.648986098734537809872118847788, 9.075329558379681298591581322983, 10.06807249695752855516836934676