L(s) = 1 | + (−0.190 + 0.587i)3-s + (0.5 + 0.363i)9-s + (−1.30 + 0.951i)17-s + (−0.5 + 1.53i)19-s + (0.309 − 0.951i)25-s + (−0.809 + 0.587i)27-s + (−0.190 + 0.587i)41-s + 0.618·43-s + (−0.809 + 0.587i)49-s + (−0.309 − 0.951i)51-s + (−0.809 − 0.587i)57-s + (0.5 + 1.53i)59-s + 1.61·67-s + (−0.190 − 0.587i)73-s + (0.5 + 0.363i)75-s + ⋯ |
L(s) = 1 | + (−0.190 + 0.587i)3-s + (0.5 + 0.363i)9-s + (−1.30 + 0.951i)17-s + (−0.5 + 1.53i)19-s + (0.309 − 0.951i)25-s + (−0.809 + 0.587i)27-s + (−0.190 + 0.587i)41-s + 0.618·43-s + (−0.809 + 0.587i)49-s + (−0.309 − 0.951i)51-s + (−0.809 − 0.587i)57-s + (0.5 + 1.53i)59-s + 1.61·67-s + (−0.190 − 0.587i)73-s + (0.5 + 0.363i)75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.001835987\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.001835987\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 5 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 - 0.618T + T^{2} \) |
| 47 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 - 1.61T + T^{2} \) |
| 71 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 - 0.618T + T^{2} \) |
| 97 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
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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
−8.821043479706943666529355907556, −8.237117546654393643851168822911, −7.50158874758530924014947005229, −6.52058393592844479030720749408, −6.00050872421252256837287407814, −5.01632901846196193911423885713, −4.25798710790906401805202790399, −3.78620995223716542567731710861, −2.46624476150023507494541106785, −1.55308601409676304603264085857,
0.57525054495186048553585225753, 1.88431614700574224108756693770, 2.73634936444670697033335533168, 3.85389014445731443461318912774, 4.70529701286478512204337184333, 5.38781381832852362317346198863, 6.63090946090857265319074495082, 6.76706738718454548832535006283, 7.50485273590827308217108484438, 8.456474080557128938633485502813