L(s) = 1 | + (0.984 − 0.173i)2-s + (−0.973 − 0.230i)3-s + (0.939 − 0.342i)4-s + (−0.448 − 0.893i)5-s + (−0.998 − 0.0581i)6-s + (−0.835 − 0.549i)7-s + (0.866 − 0.5i)8-s + (0.893 + 0.448i)9-s + (−0.597 − 0.802i)10-s + (0.597 − 0.802i)11-s + (−0.993 + 0.116i)12-s + (−0.998 − 0.0581i)13-s + (−0.918 − 0.396i)14-s + (0.230 + 0.973i)15-s + (0.766 − 0.642i)16-s + (−0.984 + 0.173i)17-s + ⋯ |
L(s) = 1 | + (0.984 − 0.173i)2-s + (−0.973 − 0.230i)3-s + (0.939 − 0.342i)4-s + (−0.448 − 0.893i)5-s + (−0.998 − 0.0581i)6-s + (−0.835 − 0.549i)7-s + (0.866 − 0.5i)8-s + (0.893 + 0.448i)9-s + (−0.597 − 0.802i)10-s + (0.597 − 0.802i)11-s + (−0.993 + 0.116i)12-s + (−0.998 − 0.0581i)13-s + (−0.918 − 0.396i)14-s + (0.230 + 0.973i)15-s + (0.766 − 0.642i)16-s + (−0.984 + 0.173i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.917 - 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.917 - 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9713741483 - 0.2016779941i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9713741483 - 0.2016779941i\) |
\(L(1)\) |
\(\approx\) |
\(0.9053158020 - 0.4514334450i\) |
\(L(1)\) |
\(\approx\) |
\(0.9053158020 - 0.4514334450i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (0.984 - 0.173i)T \) |
| 3 | \( 1 + (-0.973 - 0.230i)T \) |
| 5 | \( 1 + (-0.448 - 0.893i)T \) |
| 7 | \( 1 + (-0.835 - 0.549i)T \) |
| 11 | \( 1 + (0.597 - 0.802i)T \) |
| 13 | \( 1 + (-0.998 - 0.0581i)T \) |
| 17 | \( 1 + (-0.984 + 0.173i)T \) |
| 19 | \( 1 + iT \) |
| 23 | \( 1 + (-0.984 + 0.173i)T \) |
| 29 | \( 1 + (-0.802 + 0.597i)T \) |
| 31 | \( 1 + (0.727 - 0.686i)T \) |
| 41 | \( 1 + (-0.939 - 0.342i)T \) |
| 43 | \( 1 + (-0.984 - 0.173i)T \) |
| 47 | \( 1 + (-0.893 - 0.448i)T \) |
| 53 | \( 1 + (0.993 - 0.116i)T \) |
| 59 | \( 1 + (0.230 + 0.973i)T \) |
| 61 | \( 1 + (-0.116 - 0.993i)T \) |
| 67 | \( 1 + (0.597 - 0.802i)T \) |
| 71 | \( 1 + (0.173 - 0.984i)T \) |
| 73 | \( 1 + (0.993 + 0.116i)T \) |
| 79 | \( 1 + (-0.918 + 0.396i)T \) |
| 83 | \( 1 + (-0.0581 + 0.998i)T \) |
| 89 | \( 1 + (-0.230 + 0.973i)T \) |
| 97 | \( 1 + (-0.957 + 0.286i)T \) |
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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
−18.17539630313683714010033890791, −17.5600030498005328364621601799, −16.8873316155007785857586986680, −16.06823392178697106292800214154, −15.499440657988134786023713135258, −15.1151607312473354513042577518, −14.450390459651850486952389418878, −13.437148908112248021468671204201, −12.8014676794761007361426065584, −12.01516580836885353886841604179, −11.70187698120707257166395399203, −11.081286289821498787688876443, −10.02385438041196126640232242800, −9.79475994672326334299658879670, −8.50429678197265218954853602560, −7.29645503132022725378293822317, −6.859932100589474599380333418156, −6.46195716677125033766237480476, −5.65594770732385595364379333531, −4.72303722787794931091102554619, −4.27672718536205109432576855939, −3.39858083651976916496590124708, −2.56515407989757721124830452379, −1.86159221545751142279309153331, −0.20150919448999846817732973599,
0.48040909924535649539666375263, 1.40277678349821577737198497851, 2.21077232925828079490897121888, 3.63807249148908594608707145109, 3.89619227455935639843148912358, 4.80713101201985211018940722990, 5.42690791310441865089555652738, 6.2164262972029228954992916685, 6.73468966962734662629304514390, 7.52626048454954583931877633061, 8.30598350381206625972275981836, 9.55663787004483098840784343752, 10.09698088644300714737595173937, 10.926940142114515020418981577656, 11.65485501403465502325958689426, 12.14066765042419766350812538924, 12.67970498174005577578520898063, 13.40704449093699856660200304101, 13.80922048219861263872005921064, 14.93922077233888062546136931594, 15.6217428685885097088004388259, 16.31823477193955762169892598904, 16.82012372958187995712157690929, 17.07072977170337146245887755112, 18.36270293013693434679541975184