L(s) = 1 | + (0.841 − 0.540i)3-s + (0.959 − 0.281i)7-s + (0.415 − 0.909i)9-s + (0.654 + 0.755i)11-s + (−0.959 − 0.281i)13-s + (0.142 − 0.989i)17-s + (0.142 + 0.989i)19-s + (0.654 − 0.755i)21-s + (−0.142 − 0.989i)27-s + (0.142 − 0.989i)29-s + (0.841 + 0.540i)31-s + (0.959 + 0.281i)33-s + (0.415 − 0.909i)37-s + (−0.959 + 0.281i)39-s + (0.415 + 0.909i)41-s + ⋯ |
L(s) = 1 | + (0.841 − 0.540i)3-s + (0.959 − 0.281i)7-s + (0.415 − 0.909i)9-s + (0.654 + 0.755i)11-s + (−0.959 − 0.281i)13-s + (0.142 − 0.989i)17-s + (0.142 + 0.989i)19-s + (0.654 − 0.755i)21-s + (−0.142 − 0.989i)27-s + (0.142 − 0.989i)29-s + (0.841 + 0.540i)31-s + (0.959 + 0.281i)33-s + (0.415 − 0.909i)37-s + (−0.959 + 0.281i)39-s + (0.415 + 0.909i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.635 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.635 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.087358173 - 0.9850567605i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.087358173 - 0.9850567605i\) |
\(L(1)\) |
\(\approx\) |
\(1.528708790 - 0.3863474512i\) |
\(L(1)\) |
\(\approx\) |
\(1.528708790 - 0.3863474512i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (0.841 - 0.540i)T \) |
| 7 | \( 1 + (0.959 - 0.281i)T \) |
| 11 | \( 1 + (0.654 + 0.755i)T \) |
| 13 | \( 1 + (-0.959 - 0.281i)T \) |
| 17 | \( 1 + (0.142 - 0.989i)T \) |
| 19 | \( 1 + (0.142 + 0.989i)T \) |
| 29 | \( 1 + (0.142 - 0.989i)T \) |
| 31 | \( 1 + (0.841 + 0.540i)T \) |
| 37 | \( 1 + (0.415 - 0.909i)T \) |
| 41 | \( 1 + (0.415 + 0.909i)T \) |
| 43 | \( 1 + (0.841 - 0.540i)T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + (-0.959 + 0.281i)T \) |
| 59 | \( 1 + (0.959 + 0.281i)T \) |
| 61 | \( 1 + (-0.841 - 0.540i)T \) |
| 67 | \( 1 + (-0.654 + 0.755i)T \) |
| 71 | \( 1 + (-0.654 + 0.755i)T \) |
| 73 | \( 1 + (0.142 + 0.989i)T \) |
| 79 | \( 1 + (-0.959 - 0.281i)T \) |
| 83 | \( 1 + (0.415 - 0.909i)T \) |
| 89 | \( 1 + (0.841 - 0.540i)T \) |
| 97 | \( 1 + (-0.415 - 0.909i)T \) |
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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
−21.86474801497870127934628243215, −21.24615587353164187535377444186, −20.50459094322522013262229066384, −19.42677038476604668101565737053, −19.278335491707554995519882120443, −18.018527630116882910640803546673, −17.170542758958772330146404420663, −16.435642418226846299464033927894, −15.40778599583018127346580503629, −14.756994435427869797817998472832, −14.207856725285278069437271322140, −13.39256221890587421628878481716, −12.30843774147590066894963541838, −11.36290732552191066750549399789, −10.66049190409409281569834469550, −9.61802310797944349537130188812, −8.87854483020825412682890447350, −8.19380748941047269704468349980, −7.37302922756997024369483299182, −6.19442059254755922025941629112, −4.99056161697838324829776770406, −4.39805265924032939252494038752, −3.303578316188690573237726371063, −2.37865442856570264536102922133, −1.36617228174320900524573103931,
1.04132034427300331374314339370, 1.99104954647287218252188057157, 2.86795605554387530816153055150, 4.09519037061779428741393713528, 4.81718092129310798422793972039, 6.09491108196236120919749881847, 7.2675320467051106129607813293, 7.62970378513755154548535874309, 8.54172728829168396008697938249, 9.55917958149019308526551110396, 10.13089547718020274707650315567, 11.53270581158297446006979938424, 12.112269973181270956075507614462, 12.93715654108195515753594644393, 14.0557907313109038423958816641, 14.42153652776897388947578597351, 15.101228628884374293638842948782, 16.15047119838922199358249674739, 17.373436511447444959715471193390, 17.728500278262349235835489212203, 18.67934204077557038900158181147, 19.49281643181592478913283280939, 20.23078702876610283768723397817, 20.7751353391280672645232472593, 21.558520284373004111210404590269