| L(s) = 1 | + 1.73i·2-s − 0.999·4-s + (2.18 − 0.469i)5-s + 1.58i·7-s + 1.73i·8-s + (0.813 + 3.78i)10-s − 6.37·11-s + 0.939i·13-s − 2.74·14-s − 5·16-s + 5.04i·17-s − 4·19-s + (−2.18 + 0.469i)20-s − 11.0i·22-s − 3.46i·23-s + ⋯ |
| L(s) = 1 | + 1.22i·2-s − 0.499·4-s + (0.977 − 0.210i)5-s + 0.598i·7-s + 0.612i·8-s + (0.257 + 1.19i)10-s − 1.92·11-s + 0.260i·13-s − 0.733·14-s − 1.25·16-s + 1.22i·17-s − 0.917·19-s + (−0.488 + 0.105i)20-s − 2.35i·22-s − 0.722i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.210i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 + 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.362479356\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.362479356\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-2.18 + 0.469i)T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 1.73iT - 2T^{2} \) |
| 7 | \( 1 - 1.58iT - 7T^{2} \) |
| 11 | \( 1 + 6.37T + 11T^{2} \) |
| 13 | \( 1 - 0.939iT - 13T^{2} \) |
| 17 | \( 1 - 5.04iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 3.46iT - 23T^{2} \) |
| 31 | \( 1 - 2.37T + 31T^{2} \) |
| 37 | \( 1 - 10.0iT - 37T^{2} \) |
| 41 | \( 1 + 6.74T + 41T^{2} \) |
| 43 | \( 1 - 5.69iT - 43T^{2} \) |
| 47 | \( 1 - 5.69iT - 47T^{2} \) |
| 53 | \( 1 + 0.939iT - 53T^{2} \) |
| 59 | \( 1 - 0.744T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 8.51iT - 67T^{2} \) |
| 71 | \( 1 - 4.74T + 71T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 + 5.62T + 79T^{2} \) |
| 83 | \( 1 + 16.7iT - 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 - 6.92iT - 97T^{2} \) |
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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
−10.14420344260843115833212263566, −8.859464315198959907756663330677, −8.392356715868085465048960254930, −7.71409943742768091831827117097, −6.51092187021437559607256727027, −6.10048431651004580242421919663, −5.25606559183777277517886688200, −4.64308328006121757131520599320, −2.77137083396353360471567422121, −1.98867669896816222221939369017,
0.50637135791050198570641533203, 2.05895958927615268588247061142, 2.65395071942177938444270968474, 3.65493315569520899609632918764, 4.92019331034461521528071545333, 5.63567115529696747756206491348, 6.86927819880008607321345877625, 7.49744995121491811356411280126, 8.666165626059160603391485510426, 9.642844749200554176990629068724
