L(s) = 1 | + (−0.222 + 0.974i)5-s + (−0.900 + 0.433i)11-s + (−0.826 − 0.563i)13-s + (0.365 − 0.930i)17-s + (−0.5 + 0.866i)19-s + (0.623 − 0.781i)23-s + (−0.900 − 0.433i)25-s + (0.988 − 0.149i)29-s + (−0.5 + 0.866i)31-s + (−0.988 + 0.149i)37-s + (0.955 − 0.294i)41-s + (−0.955 − 0.294i)43-s + (−0.826 − 0.563i)47-s + (0.988 + 0.149i)53-s + (−0.222 − 0.974i)55-s + ⋯ |
L(s) = 1 | + (−0.222 + 0.974i)5-s + (−0.900 + 0.433i)11-s + (−0.826 − 0.563i)13-s + (0.365 − 0.930i)17-s + (−0.5 + 0.866i)19-s + (0.623 − 0.781i)23-s + (−0.900 − 0.433i)25-s + (0.988 − 0.149i)29-s + (−0.5 + 0.866i)31-s + (−0.988 + 0.149i)37-s + (0.955 − 0.294i)41-s + (−0.955 − 0.294i)43-s + (−0.826 − 0.563i)47-s + (0.988 + 0.149i)53-s + (−0.222 − 0.974i)55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.830 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.830 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.221842146 + 0.3721274369i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.221842146 + 0.3721274369i\) |
\(L(1)\) |
\(\approx\) |
\(0.8606524421 + 0.1288154784i\) |
\(L(1)\) |
\(\approx\) |
\(0.8606524421 + 0.1288154784i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.222 + 0.974i)T \) |
| 11 | \( 1 + (-0.900 + 0.433i)T \) |
| 13 | \( 1 + (-0.826 - 0.563i)T \) |
| 17 | \( 1 + (0.365 - 0.930i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (0.623 - 0.781i)T \) |
| 29 | \( 1 + (0.988 - 0.149i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + (-0.988 + 0.149i)T \) |
| 41 | \( 1 + (0.955 - 0.294i)T \) |
| 43 | \( 1 + (-0.955 - 0.294i)T \) |
| 47 | \( 1 + (-0.826 - 0.563i)T \) |
| 53 | \( 1 + (0.988 + 0.149i)T \) |
| 59 | \( 1 + (-0.955 - 0.294i)T \) |
| 61 | \( 1 + (-0.365 + 0.930i)T \) |
| 67 | \( 1 + (0.5 - 0.866i)T \) |
| 71 | \( 1 + (0.623 - 0.781i)T \) |
| 73 | \( 1 + (-0.0747 - 0.997i)T \) |
| 79 | \( 1 + (0.5 + 0.866i)T \) |
| 83 | \( 1 + (-0.826 + 0.563i)T \) |
| 89 | \( 1 + (0.826 - 0.563i)T \) |
| 97 | \( 1 + (0.5 - 0.866i)T \) |
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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
−19.790670147908920949558204354869, −19.440545322416027105975394697318, −18.64671012007065295797375071972, −17.55661164280577489728960606156, −17.07170377270086744261640256629, −16.32326582848209305708891242247, −15.60653878064726416568761131244, −14.92526913424302128318436854329, −13.96269350612978041398643797677, −13.07447707589999318195037150260, −12.70091783770405733899174899587, −11.76036571849582212073406483056, −11.05639273277659261082706106043, −10.11626140164680385361884215059, −9.317543826169680697807227114485, −8.558024681584854388622491061347, −7.885634699304576407529394237044, −7.0538804824366288310719983264, −5.98344421043302211138405992464, −5.136369706185165152421894335164, −4.554895366673536252997950346160, −3.556271207816605112058828260504, −2.50985948432030319619715290812, −1.51975058590595943049236612626, −0.44424424928609989393163871310,
0.47680417218371376668502521672, 1.999035649541277744255753808001, 2.78765144071095878027728864118, 3.44664340113632431221726108985, 4.68463421538496549601446984872, 5.321606871536558547171195627964, 6.41390246548691944692732821226, 7.17961469870022023884065702913, 7.763363036600371635692484731041, 8.621431380584186440853451527387, 9.817962874257532519079601183526, 10.35756969180440316490397116825, 10.904158800220534959214412122363, 12.07532471585778834449701330705, 12.447579463165114764077556874736, 13.53192112409818227842555060786, 14.30727671016785349284111625188, 14.93804469224769295911579797044, 15.56653427279778842923536299707, 16.389552142024500342426192875303, 17.25493225075161682055362068744, 18.19540159105853129618562140357, 18.43915932811567412093687211488, 19.40682664129487838623136871426, 20.00747223028210894107697371368