L(s) = 1 | + (−1.29 − 0.576i)2-s + (1.71 + 0.223i)3-s + (1.33 + 1.48i)4-s + (−2.27 − 2.27i)5-s + (−2.08 − 1.27i)6-s + 7-s + (−0.866 − 2.69i)8-s + (2.90 + 0.766i)9-s + (1.62 + 4.24i)10-s + (1.53 − 1.53i)11-s + (1.96 + 2.85i)12-s + (−3.63 − 3.63i)13-s + (−1.29 − 0.576i)14-s + (−3.39 − 4.40i)15-s + (−0.433 + 3.97i)16-s − 1.81i·17-s + ⋯ |
L(s) = 1 | + (−0.913 − 0.407i)2-s + (0.991 + 0.128i)3-s + (0.667 + 0.744i)4-s + (−1.01 − 1.01i)5-s + (−0.853 − 0.521i)6-s + 0.377·7-s + (−0.306 − 0.951i)8-s + (0.966 + 0.255i)9-s + (0.513 + 1.34i)10-s + (0.463 − 0.463i)11-s + (0.566 + 0.824i)12-s + (−1.00 − 1.00i)13-s + (−0.345 − 0.154i)14-s + (−0.876 − 1.13i)15-s + (−0.108 + 0.994i)16-s − 0.441i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.154 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.154 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.803009 - 0.687326i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.803009 - 0.687326i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.29 + 0.576i)T \) |
| 3 | \( 1 + (-1.71 - 0.223i)T \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + (2.27 + 2.27i)T + 5iT^{2} \) |
| 11 | \( 1 + (-1.53 + 1.53i)T - 11iT^{2} \) |
| 13 | \( 1 + (3.63 + 3.63i)T + 13iT^{2} \) |
| 17 | \( 1 + 1.81iT - 17T^{2} \) |
| 19 | \( 1 + (-4.98 + 4.98i)T - 19iT^{2} \) |
| 23 | \( 1 - 7.21iT - 23T^{2} \) |
| 29 | \( 1 + (-2.77 + 2.77i)T - 29iT^{2} \) |
| 31 | \( 1 + 4.14iT - 31T^{2} \) |
| 37 | \( 1 + (1.77 - 1.77i)T - 37iT^{2} \) |
| 41 | \( 1 + 0.517T + 41T^{2} \) |
| 43 | \( 1 + (4.67 + 4.67i)T + 43iT^{2} \) |
| 47 | \( 1 + 1.19T + 47T^{2} \) |
| 53 | \( 1 + (-7.94 - 7.94i)T + 53iT^{2} \) |
| 59 | \( 1 + (1.94 - 1.94i)T - 59iT^{2} \) |
| 61 | \( 1 + (-2.94 - 2.94i)T + 61iT^{2} \) |
| 67 | \( 1 + (7.85 - 7.85i)T - 67iT^{2} \) |
| 71 | \( 1 + 10.9iT - 71T^{2} \) |
| 73 | \( 1 - 15.7iT - 73T^{2} \) |
| 79 | \( 1 - 5.40iT - 79T^{2} \) |
| 83 | \( 1 + (-7.12 - 7.12i)T + 83iT^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−11.51344569236099030130729952959, −10.14625093950045287224259436029, −9.304762394390798605085333085397, −8.621072567059696430071815484558, −7.74295593311460639855729053044, −7.29152219369959406124283012369, −5.10433815794887387120719221184, −3.87388406187200601296909515362, −2.77713945947757106863149027519, −0.964988525498312238389328608034,
1.88491823279809712544854451660, 3.24806575279086507178490077831, 4.61753698761631237677646610308, 6.60164564193788629624599130446, 7.20007216123164223660077711652, 7.939047996949076877898949298948, 8.788389047799517520579214389687, 9.849202938493314400037901794278, 10.53505353396190587040445993254, 11.72685570767975706050303950192