L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−2 + 3.46i)7-s − 0.999·8-s + (1.5 − 2.59i)11-s + (−2 − 3.46i)13-s + (1.99 + 3.46i)14-s + (−0.5 + 0.866i)16-s + 3·17-s + 5·19-s + (−1.5 − 2.59i)22-s + (−3 − 5.19i)23-s − 3.99·26-s + 3.99·28-s + (3 − 5.19i)29-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.755 + 1.30i)7-s − 0.353·8-s + (0.452 − 0.783i)11-s + (−0.554 − 0.960i)13-s + (0.534 + 0.925i)14-s + (−0.125 + 0.216i)16-s + 0.727·17-s + 1.14·19-s + (−0.319 − 0.553i)22-s + (−0.625 − 1.08i)23-s − 0.784·26-s + 0.755·28-s + (0.557 − 0.964i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.602130199\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.602130199\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (2 - 3.46i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2 + 3.46i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.5 + 9.52i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.5 - 4.33i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 7T + 73T^{2} \) |
| 79 | \( 1 + (7 - 12.1i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (6 - 10.3i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + (-5.5 + 9.52i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.605848215609773402939544957879, −8.691404646559196459173988024168, −7.967605571141222859528505860505, −6.71248658222267425331884506450, −5.69838773170041315387570530160, −5.45702526952035240486692853150, −4.01939987469260157825986653599, −3.04061324049766734339289166861, −2.40078938437342417576423649212, −0.64604342364778283633835676747,
1.30698843814315406268481776280, 3.05693675681021222101574531057, 3.95731197952804616037349888435, 4.66996691897144852334544346830, 5.74680716443291661824498655398, 6.76619595709434143554395487968, 7.24627048369732391999283594904, 7.84144896634660928446929687852, 9.215091751176277948806727407592, 9.699351818900393595822257050111