L(s) = 1 | + (0.625 − 0.780i)2-s + (−0.217 − 0.976i)4-s + (−0.888 + 0.458i)7-s + (−0.897 − 0.441i)8-s + (0.290 − 0.956i)11-s + (0.272 − 0.962i)13-s + (−0.198 + 0.980i)14-s + (−0.905 + 0.424i)16-s + (0.217 − 0.976i)17-s + (−0.710 + 0.703i)19-s + (−0.564 − 0.825i)22-s + (−0.380 + 0.924i)23-s + (−0.580 − 0.814i)26-s + (0.640 + 0.768i)28-s + (0.978 + 0.207i)29-s + ⋯ |
L(s) = 1 | + (0.625 − 0.780i)2-s + (−0.217 − 0.976i)4-s + (−0.888 + 0.458i)7-s + (−0.897 − 0.441i)8-s + (0.290 − 0.956i)11-s + (0.272 − 0.962i)13-s + (−0.198 + 0.980i)14-s + (−0.905 + 0.424i)16-s + (0.217 − 0.976i)17-s + (−0.710 + 0.703i)19-s + (−0.564 − 0.825i)22-s + (−0.380 + 0.924i)23-s + (−0.580 − 0.814i)26-s + (0.640 + 0.768i)28-s + (0.978 + 0.207i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5025 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.829 - 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5025 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.829 - 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7169297235 - 2.346108706i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7169297235 - 2.346108706i\) |
\(L(1)\) |
\(\approx\) |
\(1.042832024 - 0.7230546714i\) |
\(L(1)\) |
\(\approx\) |
\(1.042832024 - 0.7230546714i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 67 | \( 1 \) |
good | 2 | \( 1 + (0.625 - 0.780i)T \) |
| 7 | \( 1 + (-0.888 + 0.458i)T \) |
| 11 | \( 1 + (0.290 - 0.956i)T \) |
| 13 | \( 1 + (0.272 - 0.962i)T \) |
| 17 | \( 1 + (0.217 - 0.976i)T \) |
| 19 | \( 1 + (-0.710 + 0.703i)T \) |
| 23 | \( 1 + (-0.380 + 0.924i)T \) |
| 29 | \( 1 + (0.978 + 0.207i)T \) |
| 31 | \( 1 + (-0.830 + 0.556i)T \) |
| 37 | \( 1 + (0.913 + 0.406i)T \) |
| 41 | \( 1 + (0.948 + 0.318i)T \) |
| 43 | \( 1 + (0.415 - 0.909i)T \) |
| 47 | \( 1 + (-0.879 - 0.475i)T \) |
| 53 | \( 1 + (0.736 - 0.676i)T \) |
| 59 | \( 1 + (0.466 + 0.884i)T \) |
| 61 | \( 1 + (0.820 - 0.572i)T \) |
| 71 | \( 1 + (0.217 + 0.976i)T \) |
| 73 | \( 1 + (0.820 - 0.572i)T \) |
| 79 | \( 1 + (-0.0665 + 0.997i)T \) |
| 83 | \( 1 + (-0.483 + 0.875i)T \) |
| 89 | \( 1 + (-0.941 - 0.336i)T \) |
| 97 | \( 1 + (0.669 - 0.743i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.87060702502952570588998052798, −17.35149319682444895601990849973, −16.605689458710588602443074699128, −16.23805005942091123645980973843, −15.48664397065928069329660607183, −14.70189114845223026754803020300, −14.34190874458044760250756770824, −13.47492156581334970937539536750, −12.74755456028945920973937107989, −12.54859148846241144463208476158, −11.577087855405696353498360354455, −10.803176075662748032158015792436, −9.89191963022917567872217227424, −9.24133108995349238192072432062, −8.57452369630089385553198440457, −7.69233263010414671287686920910, −7.052166651385366883671010022528, −6.2985614512455926068495726717, −6.120303900121273004876890188049, −4.81028533736547394412984036291, −4.20270079991150929599718221966, −3.82754209970504042069572591807, −2.71421859748277118245046464236, −2.022824999482488880967770116151, −0.67565260511499988226074805592,
0.4074704581457919511328593063, 1.03259494908742455629632787011, 2.14750649846936944375421200309, 2.951759344646414844308938444707, 3.4351920732553693678429847321, 4.113563979853491750301887705049, 5.26989674965852357968252015619, 5.69603190376130047004481620119, 6.32501662027080743204561544301, 7.13820001752409901712195395015, 8.31464540607262144043714171300, 8.84636451501016851416865879932, 9.78763572156693715425551404588, 10.09736819366452080758103171849, 11.0632120177606589858182059650, 11.54109770875333155215286233124, 12.38866224997304018492406797043, 12.811110799846128882018429492487, 13.54026859650730488162040155801, 14.10163986287648114005837514611, 14.82765240281114490491930125415, 15.61115458355214834315407503974, 16.09789689220109382352181613341, 16.79753605716737653514491539437, 18.002605574840384825804441145478