L(s) = 1 | + (0.5 − 0.866i)7-s + (0.406 + 0.913i)11-s + (−0.406 + 0.913i)13-s + (−0.309 + 0.951i)17-s + (0.951 + 0.309i)19-s + (−0.104 + 0.994i)23-s + (−0.207 + 0.978i)29-s + (−0.978 + 0.207i)31-s + (0.587 + 0.809i)37-s + (0.913 + 0.406i)41-s + (0.866 + 0.5i)43-s + (0.978 + 0.207i)47-s + (−0.5 − 0.866i)49-s + (−0.951 + 0.309i)53-s + (−0.406 + 0.913i)59-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)7-s + (0.406 + 0.913i)11-s + (−0.406 + 0.913i)13-s + (−0.309 + 0.951i)17-s + (0.951 + 0.309i)19-s + (−0.104 + 0.994i)23-s + (−0.207 + 0.978i)29-s + (−0.978 + 0.207i)31-s + (0.587 + 0.809i)37-s + (0.913 + 0.406i)41-s + (0.866 + 0.5i)43-s + (0.978 + 0.207i)47-s + (−0.5 − 0.866i)49-s + (−0.951 + 0.309i)53-s + (−0.406 + 0.913i)59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.905 + 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.905 + 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3460854047 + 1.554668662i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3460854047 + 1.554668662i\) |
\(L(1)\) |
\(\approx\) |
\(1.056034252 + 0.2511700967i\) |
\(L(1)\) |
\(\approx\) |
\(1.056034252 + 0.2511700967i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (0.5 - 0.866i)T \) |
| 11 | \( 1 + (0.406 + 0.913i)T \) |
| 13 | \( 1 + (-0.406 + 0.913i)T \) |
| 17 | \( 1 + (-0.309 + 0.951i)T \) |
| 19 | \( 1 + (0.951 + 0.309i)T \) |
| 23 | \( 1 + (-0.104 + 0.994i)T \) |
| 29 | \( 1 + (-0.207 + 0.978i)T \) |
| 31 | \( 1 + (-0.978 + 0.207i)T \) |
| 37 | \( 1 + (0.587 + 0.809i)T \) |
| 41 | \( 1 + (0.913 + 0.406i)T \) |
| 43 | \( 1 + (0.866 + 0.5i)T \) |
| 47 | \( 1 + (0.978 + 0.207i)T \) |
| 53 | \( 1 + (-0.951 + 0.309i)T \) |
| 59 | \( 1 + (-0.406 + 0.913i)T \) |
| 61 | \( 1 + (0.406 + 0.913i)T \) |
| 67 | \( 1 + (-0.207 - 0.978i)T \) |
| 71 | \( 1 + (0.309 + 0.951i)T \) |
| 73 | \( 1 + (0.809 + 0.587i)T \) |
| 79 | \( 1 + (-0.978 - 0.207i)T \) |
| 83 | \( 1 + (0.743 - 0.669i)T \) |
| 89 | \( 1 + (-0.809 - 0.587i)T \) |
| 97 | \( 1 + (-0.978 - 0.207i)T \) |
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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
−18.20888023982240116550839076188, −17.66819314513604534897305343169, −16.88369733837860786261430367966, −15.9997742048870828813245669649, −15.62244543853863699804508008078, −14.69999739527692414261041101489, −14.188197316521237359310829502924, −13.45983996321350697770970877005, −12.53935889143364310774684456772, −12.04677648679270974019071001683, −11.13297248496993160141693912557, −10.85810440422599027658971722533, −9.498261632428696478525472054460, −9.27278126865351811278878806655, −8.28770767028810126305324525958, −7.73939783460646836553051088542, −6.8824946893725120594679664652, −5.84202604140695036233367656918, −5.47698854155698314345836489987, −4.61601090629155617733648802421, −3.66693379261030057549730503311, −2.71654006514426464184320566832, −2.24010336142456516344093510278, −0.91839179259667074502684769564, −0.2618583560246445488368005413,
1.31677608892840041749242161270, 1.56156897542572440860631547314, 2.7353328970089508882679676696, 3.87308017455223406126463178699, 4.266557532239000573692271421019, 5.10950790031379453196717744449, 5.993326414577246054857876849299, 6.980578118549541436315977607658, 7.38564529845349387923864898001, 8.10235315717455835338576479484, 9.21994909870781815456940604057, 9.58730066498816554291415830416, 10.51758362986527432425713212867, 11.14515370275130049619003510069, 11.85803879435791092995001859237, 12.57736284564399561878722077128, 13.31284958889096338987936565402, 14.196496525201201261752573160944, 14.51184869082231836905216727249, 15.30244452343722908204165941151, 16.20076059519939791078135850361, 16.82696978293471593892912571365, 17.45516777978939239969251762631, 17.9755927021029473865379067012, 18.80490558026933325531611076589