L(s) = 1 | + (0.987 + 0.160i)2-s + 3-s + (0.948 + 0.316i)4-s + (−0.845 − 0.534i)5-s + (0.987 + 0.160i)6-s + (−0.632 − 0.774i)7-s + (0.885 + 0.464i)8-s + 9-s + (−0.748 − 0.663i)10-s + (0.278 − 0.960i)11-s + (0.948 + 0.316i)12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.845 − 0.534i)15-s + (0.799 + 0.600i)16-s + (0.692 − 0.721i)17-s + ⋯ |
L(s) = 1 | + (0.987 + 0.160i)2-s + 3-s + (0.948 + 0.316i)4-s + (−0.845 − 0.534i)5-s + (0.987 + 0.160i)6-s + (−0.632 − 0.774i)7-s + (0.885 + 0.464i)8-s + 9-s + (−0.748 − 0.663i)10-s + (0.278 − 0.960i)11-s + (0.948 + 0.316i)12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.845 − 0.534i)15-s + (0.799 + 0.600i)16-s + (0.692 − 0.721i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.027010257 - 0.7609092633i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.027010257 - 0.7609092633i\) |
\(L(1)\) |
\(\approx\) |
\(2.232875384 - 0.2276730097i\) |
\(L(1)\) |
\(\approx\) |
\(2.232875384 - 0.2276730097i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 547 | \( 1 \) |
good | 2 | \( 1 + (0.987 + 0.160i)T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + (-0.845 - 0.534i)T \) |
| 7 | \( 1 + (-0.632 - 0.774i)T \) |
| 11 | \( 1 + (0.278 - 0.960i)T \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
| 17 | \( 1 + (0.692 - 0.721i)T \) |
| 19 | \( 1 + (0.799 - 0.600i)T \) |
| 23 | \( 1 + (-0.996 - 0.0804i)T \) |
| 29 | \( 1 + (0.568 + 0.822i)T \) |
| 31 | \( 1 + (0.120 - 0.992i)T \) |
| 37 | \( 1 + (-0.996 + 0.0804i)T \) |
| 41 | \( 1 + (-0.5 + 0.866i)T \) |
| 43 | \( 1 + (-0.200 - 0.979i)T \) |
| 47 | \( 1 + (0.428 - 0.903i)T \) |
| 53 | \( 1 + (-0.0402 + 0.999i)T \) |
| 59 | \( 1 + (0.799 + 0.600i)T \) |
| 61 | \( 1 + (0.987 + 0.160i)T \) |
| 67 | \( 1 + (-0.200 - 0.979i)T \) |
| 71 | \( 1 + (-0.919 - 0.391i)T \) |
| 73 | \( 1 + (-0.845 + 0.534i)T \) |
| 79 | \( 1 + (-0.748 + 0.663i)T \) |
| 83 | \( 1 + (-0.0402 + 0.999i)T \) |
| 89 | \( 1 + (-0.354 + 0.935i)T \) |
| 97 | \( 1 + (0.278 + 0.960i)T \) |
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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
−23.29741254480813038433086304684, −22.50229369111472797061366868476, −21.97792006386328377475651139114, −20.87641615692699341509532835474, −20.118928308109537735609627218978, −19.41679173740826264367322495260, −18.92431414613940664558741718742, −17.698150900911371169514531998249, −15.88470649485484459466834029127, −15.79145174349898577642458077636, −14.658558476092710352214715528815, −14.45508816909709631361160856003, −13.12094264527696505118804427280, −12.278636732279521367163038441116, −11.92887964781896101554580772576, −10.24148477228007775926299271804, −9.9321824820907348475619381431, −8.35158962930287170760129679289, −7.54497970189947105390461821357, −6.73443942809781843272198493725, −5.57229262632571492642131632621, −4.332247744509195134609865204100, −3.43835162691268893415059394422, −2.83254899071939027763355736540, −1.739096839643202216597118896092,
1.173214224217682735203477666572, 2.7319622387233310879323180671, 3.591858307098225876269548689768, 4.186049804410975996921586495353, 5.249209556594237128089530623620, 6.75656842145784576827526471420, 7.34940120588115040429540279617, 8.24634931106625310450163038348, 9.2832796156247963543616593518, 10.36771138759334660016201753965, 11.6565024709414267307993169970, 12.220026884267741368606355096621, 13.41056958290185513761830624131, 13.81991093763815095715334307489, 14.62548740824973857488668119587, 15.746380737455590186098883641098, 16.23934362281195677426753325742, 16.89493445246961085816211813692, 18.702711291148162436842487443308, 19.49952446930996000485207194762, 20.074211184173109452457892645195, 20.681392639203725926037147831578, 21.701146991782810124128642478169, 22.4251018050196782673713317191, 23.59486671910174905398532892431