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
−10.03498125126303768559099391071, −8.689223489429598187327129036659, −8.129033338426144666038968600108, −7.13239069102917032161630753226, −6.39911864534848620749143303972, −5.18562359408352725886083039487, −4.75921206485804303885288800752, −3.57734765278609824741732005798, −2.55998803375001182095488532862, −1.14538182954629501756742180596,
1.55162904800395650213867468481, 2.54062012141973695952783034456, 3.96919643955467134984066752511, 4.56813176005885191995840177043, 5.57783172556145096946663709445, 6.44499856994458284274529736581, 7.20044604348840720991837768708, 8.433095102713595573631883100319, 8.747743310230295735259609702422, 10.02970648744646252098889400772