L(s) = 1 | + (−0.0784 − 0.996i)3-s + (0.453 − 0.891i)7-s + (−0.987 + 0.156i)9-s + (−0.233 + 0.972i)13-s + (−0.587 − 0.809i)17-s + (−0.996 + 0.0784i)19-s + (−0.923 − 0.382i)21-s + (0.707 − 0.707i)23-s + (0.233 + 0.972i)27-s + (0.649 + 0.760i)29-s + (0.809 + 0.587i)31-s + (0.996 + 0.0784i)37-s + (0.987 + 0.156i)39-s + (0.453 + 0.891i)41-s + (0.923 + 0.382i)43-s + ⋯ |
L(s) = 1 | + (−0.0784 − 0.996i)3-s + (0.453 − 0.891i)7-s + (−0.987 + 0.156i)9-s + (−0.233 + 0.972i)13-s + (−0.587 − 0.809i)17-s + (−0.996 + 0.0784i)19-s + (−0.923 − 0.382i)21-s + (0.707 − 0.707i)23-s + (0.233 + 0.972i)27-s + (0.649 + 0.760i)29-s + (0.809 + 0.587i)31-s + (0.996 + 0.0784i)37-s + (0.987 + 0.156i)39-s + (0.453 + 0.891i)41-s + (0.923 + 0.382i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0131 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0131 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.115061794 - 1.129816741i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.115061794 - 1.129816741i\) |
\(L(1)\) |
\(\approx\) |
\(0.9683383815 - 0.4269458418i\) |
\(L(1)\) |
\(\approx\) |
\(0.9683383815 - 0.4269458418i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + (-0.0784 - 0.996i)T \) |
| 7 | \( 1 + (0.453 - 0.891i)T \) |
| 13 | \( 1 + (-0.233 + 0.972i)T \) |
| 17 | \( 1 + (-0.587 - 0.809i)T \) |
| 19 | \( 1 + (-0.996 + 0.0784i)T \) |
| 23 | \( 1 + (0.707 - 0.707i)T \) |
| 29 | \( 1 + (0.649 + 0.760i)T \) |
| 31 | \( 1 + (0.809 + 0.587i)T \) |
| 37 | \( 1 + (0.996 + 0.0784i)T \) |
| 41 | \( 1 + (0.453 + 0.891i)T \) |
| 43 | \( 1 + (0.923 + 0.382i)T \) |
| 47 | \( 1 + (0.951 + 0.309i)T \) |
| 53 | \( 1 + (0.522 + 0.852i)T \) |
| 59 | \( 1 + (0.996 + 0.0784i)T \) |
| 61 | \( 1 + (0.522 - 0.852i)T \) |
| 67 | \( 1 + (0.923 - 0.382i)T \) |
| 71 | \( 1 + (-0.987 - 0.156i)T \) |
| 73 | \( 1 + (-0.891 - 0.453i)T \) |
| 79 | \( 1 + (0.587 - 0.809i)T \) |
| 83 | \( 1 + (-0.852 - 0.522i)T \) |
| 89 | \( 1 + (-0.707 - 0.707i)T \) |
| 97 | \( 1 + (-0.809 - 0.587i)T \) |
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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
−19.16785127072169536241192138269, −17.980624714678514125231382497114, −17.41341465424279160747535604131, −17.03837588639546310738731796212, −15.912007877353453278448627125086, −15.39519488797885935289246911961, −15.01019336398094237385832170883, −14.360278854853934098062063426830, −13.32318446713308460281609375301, −12.65483339087021213326215233028, −11.79953861217518534350349919353, −11.19180896871258127561928978908, −10.50627940777732498366382429538, −9.87773401767310858584974064877, −8.99872936403497477231858349227, −8.48248602236066340487713190816, −7.846039184135940610652010410685, −6.64940634929888190688945891381, −5.729392581779130441280597956699, −5.41664244379277436576792686844, −4.374596579294348066655960915930, −3.90445938124727768740835869240, −2.64849509051088288217223407163, −2.34936138438730378892273043008, −0.824467601197573682144763563192,
0.62894069327265499872132764393, 1.39741352821707095085828607265, 2.3331175290976552409723782447, 2.997094139905684504730455389, 4.394700316801307210465942972731, 4.63341550629323044093949788277, 5.8457137929073544938965338104, 6.75161609238268484281870725813, 6.99157565858696625956511033506, 7.868674391288950770774759199765, 8.60016001123921809356828759434, 9.245780454156788723882775403603, 10.34425050738940283021514167875, 11.04434504008326602691454011545, 11.558266079273804016158319634651, 12.39937799059283359926600274096, 13.03247539044449593127412550650, 13.749701184075704975987586084591, 14.30707112740632488534878536762, 14.81564188967492927451929818703, 16.02454332798721847424252966066, 16.66186319044825139579243180886, 17.28642008463564186927946771312, 17.84269357467840844038777775963, 18.5718435507620020302600741272