L(s) = 1 | + (0.951 − 0.309i)3-s + 7-s + (0.809 − 0.587i)9-s + (0.587 − 0.809i)11-s + (−0.587 − 0.809i)13-s + (0.309 − 0.951i)17-s + (−0.951 − 0.309i)19-s + (0.951 − 0.309i)21-s + (−0.809 − 0.587i)23-s + (0.587 − 0.809i)27-s + (−0.951 + 0.309i)29-s + (−0.309 + 0.951i)31-s + (0.309 − 0.951i)33-s + (0.587 + 0.809i)37-s + (−0.809 − 0.587i)39-s + ⋯ |
L(s) = 1 | + (0.951 − 0.309i)3-s + 7-s + (0.809 − 0.587i)9-s + (0.587 − 0.809i)11-s + (−0.587 − 0.809i)13-s + (0.309 − 0.951i)17-s + (−0.951 − 0.309i)19-s + (0.951 − 0.309i)21-s + (−0.809 − 0.587i)23-s + (0.587 − 0.809i)27-s + (−0.951 + 0.309i)29-s + (−0.309 + 0.951i)31-s + (0.309 − 0.951i)33-s + (0.587 + 0.809i)37-s + (−0.809 − 0.587i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0157 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0157 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.159830930 - 2.126168080i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.159830930 - 2.126168080i\) |
\(L(1)\) |
\(\approx\) |
\(1.545633023 - 0.5270622424i\) |
\(L(1)\) |
\(\approx\) |
\(1.545633023 - 0.5270622424i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.951 - 0.309i)T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + (0.587 - 0.809i)T \) |
| 13 | \( 1 + (-0.587 - 0.809i)T \) |
| 17 | \( 1 + (0.309 - 0.951i)T \) |
| 19 | \( 1 + (-0.951 - 0.309i)T \) |
| 23 | \( 1 + (-0.809 - 0.587i)T \) |
| 29 | \( 1 + (-0.951 + 0.309i)T \) |
| 31 | \( 1 + (-0.309 + 0.951i)T \) |
| 37 | \( 1 + (0.587 + 0.809i)T \) |
| 41 | \( 1 + (0.809 - 0.587i)T \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 + (-0.309 - 0.951i)T \) |
| 53 | \( 1 + (0.951 - 0.309i)T \) |
| 59 | \( 1 + (-0.587 - 0.809i)T \) |
| 61 | \( 1 + (0.587 - 0.809i)T \) |
| 67 | \( 1 + (-0.951 - 0.309i)T \) |
| 71 | \( 1 + (0.309 + 0.951i)T \) |
| 73 | \( 1 + (0.809 + 0.587i)T \) |
| 79 | \( 1 + (-0.309 - 0.951i)T \) |
| 83 | \( 1 + (-0.951 - 0.309i)T \) |
| 89 | \( 1 + (0.809 + 0.587i)T \) |
| 97 | \( 1 + (0.309 + 0.951i)T \) |
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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
−24.3363303304238936785083071135, −23.8238543772359735918303960670, −22.486916682307087272837295883846, −21.489373768520220067145109911964, −21.02593517955189534859696457073, −19.99115584384215571019015432857, −19.36237492483365216355603509374, −18.37718021213622291339049816133, −17.292697833108205718599977048113, −16.54544090326179298720725464364, −15.11510092393418433691981848590, −14.778862899200605657710957818674, −13.99885474599928799493194025335, −12.864003236591440142124331608200, −11.89207435559984608418787412624, −10.79341617499017525001034459724, −9.76611221198029227717454882550, −8.98998684100095395804239835197, −7.94213929495409554573655234971, −7.27809308876898137110014065608, −5.827445170795368911673034378360, −4.35657740067398569165091540382, −4.01020364454513864065771350617, −2.23996540652591538208077412418, −1.65672478104449074294656478037,
0.70747467970632277382970154844, 1.9559425704431406981900670989, 2.99956877966409922307690605063, 4.139612901269590021528959052510, 5.270188817401354209168297673351, 6.61127343713475389113063758960, 7.67440780122434456708113078736, 8.37935551365406467633154836771, 9.23424020505404350106711598469, 10.36916283306375115820143975660, 11.44615018231584335411117838944, 12.42517007829211874869307278008, 13.40025408657631409102321851322, 14.398792091395184018976303685781, 14.74052239226812485701252946197, 15.90304360334993660734854098856, 17.01155373011810948713964570256, 18.01902939142013469045971638532, 18.70127276084232381723662656285, 19.75692505047777792633134668617, 20.33775410972734843426625081883, 21.27957936639735581556501137766, 22.00371682019542090027502940376, 23.24891225809609155835847284169, 24.323467429793120585480302507581