L(s) = 1 | + (0.0605 − 0.998i)2-s + (−0.992 − 0.120i)4-s + (−0.754 + 0.656i)5-s + (−0.180 + 0.983i)8-s + (0.609 + 0.792i)10-s + (0.962 − 0.272i)11-s + (0.627 − 0.778i)13-s + (0.970 + 0.240i)16-s + (−0.126 − 0.991i)17-s + (0.609 − 0.792i)19-s + (0.828 − 0.560i)20-s + (−0.213 − 0.976i)22-s + (−0.565 − 0.824i)23-s + (0.137 − 0.990i)25-s + (−0.739 − 0.673i)26-s + ⋯ |
L(s) = 1 | + (0.0605 − 0.998i)2-s + (−0.992 − 0.120i)4-s + (−0.754 + 0.656i)5-s + (−0.180 + 0.983i)8-s + (0.609 + 0.792i)10-s + (0.962 − 0.272i)11-s + (0.627 − 0.778i)13-s + (0.970 + 0.240i)16-s + (−0.126 − 0.991i)17-s + (0.609 − 0.792i)19-s + (0.828 − 0.560i)20-s + (−0.213 − 0.976i)22-s + (−0.565 − 0.824i)23-s + (0.137 − 0.990i)25-s + (−0.739 − 0.673i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.897 - 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.897 - 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2771541916 - 1.189623716i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2771541916 - 1.189623716i\) |
\(L(1)\) |
\(\approx\) |
\(0.7645448446 - 0.5164787391i\) |
\(L(1)\) |
\(\approx\) |
\(0.7645448446 - 0.5164787391i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 191 | \( 1 \) |
good | 2 | \( 1 + (0.0605 - 0.998i)T \) |
| 5 | \( 1 + (-0.754 + 0.656i)T \) |
| 11 | \( 1 + (0.962 - 0.272i)T \) |
| 13 | \( 1 + (0.627 - 0.778i)T \) |
| 17 | \( 1 + (-0.126 - 0.991i)T \) |
| 19 | \( 1 + (0.609 - 0.792i)T \) |
| 23 | \( 1 + (-0.565 - 0.824i)T \) |
| 29 | \( 1 + (0.340 + 0.940i)T \) |
| 31 | \( 1 + (0.592 - 0.805i)T \) |
| 37 | \( 1 + (-0.592 - 0.805i)T \) |
| 41 | \( 1 + (-0.986 + 0.164i)T \) |
| 43 | \( 1 + (0.490 + 0.871i)T \) |
| 47 | \( 1 + (0.884 - 0.466i)T \) |
| 53 | \( 1 + (-0.618 + 0.785i)T \) |
| 59 | \( 1 + (0.00551 - 0.999i)T \) |
| 61 | \( 1 + (0.480 + 0.876i)T \) |
| 67 | \( 1 + (0.287 + 0.957i)T \) |
| 71 | \( 1 + (0.148 - 0.988i)T \) |
| 73 | \( 1 + (0.441 + 0.897i)T \) |
| 79 | \( 1 + (-0.644 - 0.764i)T \) |
| 83 | \( 1 + (0.431 + 0.901i)T \) |
| 89 | \( 1 + (0.0935 - 0.995i)T \) |
| 97 | \( 1 + (0.995 + 0.0990i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.96556187469785238384636021664, −17.83406603511048140155242836465, −17.18665107729757753513256769643, −16.76134669876119871860003245923, −15.926643305256307081393937918685, −15.55564931344987056715829501925, −14.83384801282455292291842900417, −13.92832254581434599553293351380, −13.64642957623048567622023235596, −12.53545445376650408763927933423, −12.08416196009319265692385029006, −11.449272070719235311636346693375, −10.27404514384588471421935606211, −9.54339831040967654993735722423, −8.80959946096582779264314312968, −8.3029368273722427974208688671, −7.64391293150462221830850080678, −6.786406556156148557962380928427, −6.2069299649311935648487208517, −5.37943918137651223521300497386, −4.547453738154724326212844332976, −3.85444025334374024681931643655, −3.51022945410677948382294939300, −1.7267447201444451119304702109, −1.050378319445445558881418778873,
0.4348578373564437446158423705, 1.1815387942275184516043873588, 2.44174676840519262508671287604, 3.03393619761930401207699363405, 3.71609791142657112623367712806, 4.39704717481123923565511338437, 5.23016230588942579096178874543, 6.18379337731371967876999561831, 6.97887257043175640546659035141, 7.841696673251602027788751564233, 8.597142452967342687900876520983, 9.20418312408541430103573932799, 10.078823335102123413992742576245, 10.74759922333268825943274833364, 11.363015652940491110931327545470, 11.84566665780232329431406479208, 12.51744024958683622555328864828, 13.39116921216145981818695690911, 14.09221348008929495655098661493, 14.52087604231066079416372220454, 15.474172142641761510820602554017, 16.02386579453238641929205923751, 16.97657459301744449534417758044, 17.88038193493244903900470818807, 18.24727569819102026821521127309