L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)7-s + 0.999·8-s + (2 + 3.46i)13-s + (−0.499 − 0.866i)14-s + (−0.5 + 0.866i)16-s − 6·17-s + 2·19-s + (2.5 − 4.33i)25-s − 3.99·26-s + 0.999·28-s + (−3 + 5.19i)29-s + (2 + 3.46i)31-s + (−0.499 − 0.866i)32-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.188 + 0.327i)7-s + 0.353·8-s + (0.554 + 0.960i)13-s + (−0.133 − 0.231i)14-s + (−0.125 + 0.216i)16-s − 1.45·17-s + 0.458·19-s + (0.5 − 0.866i)25-s − 0.784·26-s + 0.188·28-s + (−0.557 + 0.964i)29-s + (0.359 + 0.622i)31-s + (−0.0883 − 0.153i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8836608264\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8836608264\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
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 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (-3 - 5.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (6 - 10.3i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3 - 5.19i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + (-5 + 8.66i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.891313454223295772593032602760, −9.144112326093391275416968878880, −8.608130871073625653206077853633, −7.69057115866209479965529724330, −6.58825241191196751708728243688, −6.31892139665696847393349166371, −5.01278522753931608490796923940, −4.26342269721344477796631016364, −2.87292493686573118849481064939, −1.48752664366294357642501161569,
0.44535638382632330893378840750, 1.94393534337681166868019465276, 3.13956001154375328248567534051, 4.01208733707856319603326521253, 5.08966268764508194729879453026, 6.16434002775949701438081814186, 7.15019160203012861630806528079, 7.981257120415105270255681726485, 8.808177010598245969218362258317, 9.535472416805858380599696163061