L(s) = 1 | + (−0.976 + 0.214i)2-s + (0.907 − 0.419i)4-s + (−0.267 + 0.963i)5-s + (−0.725 + 0.687i)7-s + (−0.796 + 0.605i)8-s + (0.0541 − 0.998i)10-s + (−0.647 − 0.762i)11-s + (−0.947 + 0.319i)13-s + (0.561 − 0.827i)14-s + (0.647 − 0.762i)16-s + (0.725 + 0.687i)17-s + (−0.370 − 0.928i)19-s + (0.161 + 0.986i)20-s + (0.796 + 0.605i)22-s + (0.994 − 0.108i)23-s + ⋯ |
L(s) = 1 | + (−0.976 + 0.214i)2-s + (0.907 − 0.419i)4-s + (−0.267 + 0.963i)5-s + (−0.725 + 0.687i)7-s + (−0.796 + 0.605i)8-s + (0.0541 − 0.998i)10-s + (−0.647 − 0.762i)11-s + (−0.947 + 0.319i)13-s + (0.561 − 0.827i)14-s + (0.647 − 0.762i)16-s + (0.725 + 0.687i)17-s + (−0.370 − 0.928i)19-s + (0.161 + 0.986i)20-s + (0.796 + 0.605i)22-s + (0.994 − 0.108i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.324 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.324 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2790386478 - 0.1992912974i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2790386478 - 0.1992912974i\) |
\(L(1)\) |
\(\approx\) |
\(0.5048316767 + 0.1060353904i\) |
\(L(1)\) |
\(\approx\) |
\(0.5048316767 + 0.1060353904i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 59 | \( 1 \) |
good | 2 | \( 1 + (-0.976 + 0.214i)T \) |
| 5 | \( 1 + (-0.267 + 0.963i)T \) |
| 7 | \( 1 + (-0.725 + 0.687i)T \) |
| 11 | \( 1 + (-0.647 - 0.762i)T \) |
| 13 | \( 1 + (-0.947 + 0.319i)T \) |
| 17 | \( 1 + (0.725 + 0.687i)T \) |
| 19 | \( 1 + (-0.370 - 0.928i)T \) |
| 23 | \( 1 + (0.994 - 0.108i)T \) |
| 29 | \( 1 + (-0.976 - 0.214i)T \) |
| 31 | \( 1 + (-0.370 + 0.928i)T \) |
| 37 | \( 1 + (0.796 + 0.605i)T \) |
| 41 | \( 1 + (0.994 + 0.108i)T \) |
| 43 | \( 1 + (0.647 - 0.762i)T \) |
| 47 | \( 1 + (-0.267 - 0.963i)T \) |
| 53 | \( 1 + (-0.0541 - 0.998i)T \) |
| 61 | \( 1 + (0.976 - 0.214i)T \) |
| 67 | \( 1 + (0.796 - 0.605i)T \) |
| 71 | \( 1 + (-0.267 - 0.963i)T \) |
| 73 | \( 1 + (-0.561 + 0.827i)T \) |
| 79 | \( 1 + (-0.161 - 0.986i)T \) |
| 83 | \( 1 + (-0.468 - 0.883i)T \) |
| 89 | \( 1 + (-0.976 - 0.214i)T \) |
| 97 | \( 1 + (-0.561 - 0.827i)T \) |
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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
−27.43001199173083707453870603980, −26.52199234526357289503660161781, −25.4950638706292895458985587186, −24.7687826926728800475157764071, −23.61300856578236745044611191382, −22.63145010849241701512626361124, −21.03181027559844559146783628787, −20.458851093045230101675399388983, −19.609735481322490416772647436140, −18.71832141051004104276453271622, −17.4174815092865073254912725029, −16.69303736421210522219513280666, −15.96177236752247099805166747466, −14.7595463600124081381628334792, −12.88645117870991250840333129591, −12.51311942286220992871978164013, −11.12893719155170488749084059338, −9.87997188631503922928073525364, −9.36224275757891009913031313698, −7.81367349600835803597377503091, −7.30951937224293620353740655779, −5.62275287446514629462243311444, −4.10649325895933709452072538680, −2.632946019209970441519053866263, −1.01335659255534354784310977720,
0.19872000300681482267629863163, 2.335599855290623458780603174647, 3.22730666533434033764474709390, 5.474381651345547010969926150618, 6.55829419217962331486487794342, 7.46851624602290861662222153930, 8.661573549074372900278945206431, 9.73886173185334656910827805224, 10.710649686525685969452256741720, 11.61829264643758869074603926804, 12.88033183583618557910856013715, 14.54010205559738684214912206705, 15.25492786197939215004924096651, 16.20528214211392337149247710165, 17.226422699083449529113685613520, 18.43084860983335251426504184155, 19.06839371945690446598500674101, 19.64503701585230258401855175637, 21.24622625107934831300519777598, 22.0160910873095232024767017186, 23.31362199050619266355195862162, 24.22947496715744904555533106564, 25.379367238822299172963896795525, 26.13807574264683353188008934294, 26.798522520302675915547155000040