L(s) = 1 | + 2-s + 4-s + (2.5 + 0.866i)7-s + 8-s + (2 + 3.46i)13-s + (2.5 + 0.866i)14-s + 16-s + (−3 + 5.19i)17-s + (−1 − 1.73i)19-s + (1.5 − 2.59i)23-s + (2.5 + 4.33i)25-s + (2 + 3.46i)26-s + (2.5 + 0.866i)28-s + (3 − 5.19i)29-s + 5·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + (0.944 + 0.327i)7-s + 0.353·8-s + (0.554 + 0.960i)13-s + (0.668 + 0.231i)14-s + 0.250·16-s + (−0.727 + 1.26i)17-s + (−0.229 − 0.397i)19-s + (0.312 − 0.541i)23-s + (0.5 + 0.866i)25-s + (0.392 + 0.679i)26-s + (0.472 + 0.163i)28-s + (0.557 − 0.964i)29-s + 0.898·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.836936757\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.836936757\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (-2.5 - 0.866i)T \) |
good | 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2 - 3.46i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1 - 1.73i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 15T + 71T^{2} \) |
| 73 | \( 1 + (5.5 - 9.52i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + T + 79T^{2} \) |
| 83 | \( 1 + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (4.5 + 7.79i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.02970648744646252098889400772, −8.747743310230295735259609702422, −8.433095102713595573631883100319, −7.20044604348840720991837768708, −6.44499856994458284274529736581, −5.57783172556145096946663709445, −4.56813176005885191995840177043, −3.96919643955467134984066752511, −2.54062012141973695952783034456, −1.55162904800395650213867468481,
1.14538182954629501756742180596, 2.55998803375001182095488532862, 3.57734765278609824741732005798, 4.75921206485804303885288800752, 5.18562359408352725886083039487, 6.39911864534848620749143303972, 7.13239069102917032161630753226, 8.129033338426144666038968600108, 8.689223489429598187327129036659, 10.03498125126303768559099391071