L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.993 + 0.116i)3-s + (0.766 + 0.642i)4-s + (−0.957 − 0.286i)5-s + (−0.893 − 0.448i)6-s + (−0.686 − 0.727i)7-s + (−0.5 − 0.866i)8-s + (0.973 + 0.230i)9-s + (0.802 + 0.597i)10-s + (−0.802 + 0.597i)11-s + (0.686 + 0.727i)12-s + (−0.686 − 0.727i)13-s + (0.396 + 0.918i)14-s + (−0.918 − 0.396i)15-s + (0.173 + 0.984i)16-s + (−0.173 − 0.984i)17-s + ⋯ |
L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.993 + 0.116i)3-s + (0.766 + 0.642i)4-s + (−0.957 − 0.286i)5-s + (−0.893 − 0.448i)6-s + (−0.686 − 0.727i)7-s + (−0.5 − 0.866i)8-s + (0.973 + 0.230i)9-s + (0.802 + 0.597i)10-s + (−0.802 + 0.597i)11-s + (0.686 + 0.727i)12-s + (−0.686 − 0.727i)13-s + (0.396 + 0.918i)14-s + (−0.918 − 0.396i)15-s + (0.173 + 0.984i)16-s + (−0.173 − 0.984i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.05929960653 - 0.5133527269i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.05929960653 - 0.5133527269i\) |
\(L(1)\) |
\(\approx\) |
\(0.6456679858 - 0.2037124030i\) |
\(L(1)\) |
\(\approx\) |
\(0.6456679858 - 0.2037124030i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (-0.939 - 0.342i)T \) |
| 3 | \( 1 + (0.993 + 0.116i)T \) |
| 5 | \( 1 + (-0.957 - 0.286i)T \) |
| 7 | \( 1 + (-0.686 - 0.727i)T \) |
| 11 | \( 1 + (-0.802 + 0.597i)T \) |
| 13 | \( 1 + (-0.686 - 0.727i)T \) |
| 17 | \( 1 + (-0.173 - 0.984i)T \) |
| 19 | \( 1 + (0.173 + 0.984i)T \) |
| 23 | \( 1 + (0.939 - 0.342i)T \) |
| 29 | \( 1 + (0.116 - 0.993i)T \) |
| 31 | \( 1 + (-0.230 + 0.973i)T \) |
| 41 | \( 1 - iT \) |
| 43 | \( 1 + (0.984 - 0.173i)T \) |
| 47 | \( 1 + (-0.998 - 0.0581i)T \) |
| 53 | \( 1 + (0.230 + 0.973i)T \) |
| 59 | \( 1 + (0.396 - 0.918i)T \) |
| 61 | \( 1 + (-0.549 - 0.835i)T \) |
| 67 | \( 1 + (0.957 - 0.286i)T \) |
| 71 | \( 1 + (0.939 - 0.342i)T \) |
| 73 | \( 1 + (-0.597 - 0.802i)T \) |
| 79 | \( 1 + (-0.686 + 0.727i)T \) |
| 83 | \( 1 + (0.993 + 0.116i)T \) |
| 89 | \( 1 + (0.549 + 0.835i)T \) |
| 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
−18.88888263037963707195528048779, −18.38001554351464340263975915859, −17.58091343176437582685361698827, −16.4374789615753159867752548725, −16.12524114166691039416517640585, −15.31321221815219263493678915710, −14.99172960747568741351634541923, −14.3766199989008483552407300941, −13.23119843705190828204335654333, −12.72838816209156811265695508244, −11.758044270080710379092258257144, −11.1347918043959252868827146889, −10.353073751457323050501739808601, −9.53341611342798790734087226443, −8.960537366836412768464703889021, −8.40004357726405441003488100628, −7.743306597211993464690785216937, −7.020891854435737773363398033085, −6.55778692197112228768328565889, −5.48605079501721972883815878956, −4.54340476269529840714352086889, −3.44040839634096623127493795220, −2.785055707916513810697933202377, −2.259789183025626602878144064865, −1.06244563128141430322250773840,
0.2049744849868935383856843393, 1.102955926381855324678167762490, 2.2856058685145533395468780554, 2.97063142235214484458097701756, 3.53961435316904555099921311119, 4.35048147219936200856209025459, 5.20038565737373568467875796122, 6.74289064745608383623320477295, 7.26742943149937242308328649208, 7.80920140009429602145446675065, 8.2703921617786962832702035483, 9.29202810396633564743628077775, 9.68767933409311342087674252395, 10.4853581392265763269165685305, 10.92566272290476703471156793278, 12.19909123168007132967672682259, 12.51748990303728185607097175995, 13.17600652078289573854618957574, 14.07717493193575244683805096255, 15.03275899005522342280439961348, 15.6066281273327423555535529197, 16.05720719998085315835418787559, 16.72694139845865906823103345013, 17.5362857412556673332225882586, 18.38623387847730589110176681454