L(s) = 1 | + (0.910 − 0.412i)3-s + (−0.0327 + 0.999i)5-s + (0.659 − 0.751i)9-s + (−0.986 − 0.162i)11-s + (−0.471 + 0.881i)13-s + (0.382 + 0.923i)15-s + (0.991 − 0.130i)17-s + (−0.973 − 0.227i)19-s + (0.0654 + 0.997i)23-s + (−0.997 − 0.0654i)25-s + (0.290 − 0.956i)27-s + (0.773 − 0.634i)29-s + (−0.965 − 0.258i)31-s + (−0.965 + 0.258i)33-s + (0.729 − 0.683i)37-s + ⋯ |
L(s) = 1 | + (0.910 − 0.412i)3-s + (−0.0327 + 0.999i)5-s + (0.659 − 0.751i)9-s + (−0.986 − 0.162i)11-s + (−0.471 + 0.881i)13-s + (0.382 + 0.923i)15-s + (0.991 − 0.130i)17-s + (−0.973 − 0.227i)19-s + (0.0654 + 0.997i)23-s + (−0.997 − 0.0654i)25-s + (0.290 − 0.956i)27-s + (0.773 − 0.634i)29-s + (−0.965 − 0.258i)31-s + (−0.965 + 0.258i)33-s + (0.729 − 0.683i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.057956453 - 1.051840477i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.057956453 - 1.051840477i\) |
\(L(1)\) |
\(\approx\) |
\(1.301238643 - 0.05741731322i\) |
\(L(1)\) |
\(\approx\) |
\(1.301238643 - 0.05741731322i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.910 - 0.412i)T \) |
| 5 | \( 1 + (-0.0327 + 0.999i)T \) |
| 11 | \( 1 + (-0.986 - 0.162i)T \) |
| 13 | \( 1 + (-0.471 + 0.881i)T \) |
| 17 | \( 1 + (0.991 - 0.130i)T \) |
| 19 | \( 1 + (-0.973 - 0.227i)T \) |
| 23 | \( 1 + (0.0654 + 0.997i)T \) |
| 29 | \( 1 + (0.773 - 0.634i)T \) |
| 31 | \( 1 + (-0.965 - 0.258i)T \) |
| 37 | \( 1 + (0.729 - 0.683i)T \) |
| 41 | \( 1 + (-0.555 - 0.831i)T \) |
| 43 | \( 1 + (-0.0980 - 0.995i)T \) |
| 47 | \( 1 + (-0.793 + 0.608i)T \) |
| 53 | \( 1 + (0.162 - 0.986i)T \) |
| 59 | \( 1 + (-0.528 + 0.849i)T \) |
| 61 | \( 1 + (0.582 - 0.812i)T \) |
| 67 | \( 1 + (0.412 + 0.910i)T \) |
| 71 | \( 1 + (0.980 - 0.195i)T \) |
| 73 | \( 1 + (0.946 - 0.321i)T \) |
| 79 | \( 1 + (-0.130 + 0.991i)T \) |
| 83 | \( 1 + (0.956 - 0.290i)T \) |
| 89 | \( 1 + (0.896 - 0.442i)T \) |
| 97 | \( 1 + (-0.707 + 0.707i)T \) |
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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
−20.06917241959836171453966226446, −19.74204278012659392136292619575, −18.69080646802101870727789621730, −18.0851141427317476522408942950, −16.89095654413436214855598276996, −16.47361419119477566915304910278, −15.65632170669816586638046786533, −14.940165681940948710930244719453, −14.36431875536051215395572759593, −13.275515103143447466961712279882, −12.7972286772164547490707547091, −12.25440197956747813901221150935, −10.87570857427728132382807332335, −10.174301504986302326952272832611, −9.62332302169912872080154605870, −8.56876575050755065559625363614, −8.139130389961895292321537924368, −7.524108654513367442096677008956, −6.22923614074193317531600292277, −5.02131808156071686608287264864, −4.81689487374834152726784591493, −3.64252564057726700106474013486, −2.82815005414844326995049690563, −1.936762316165115388070880317832, −0.83371774517585661915460716059,
0.41111671047298452555123302266, 1.90410582808267817420992504133, 2.422232151851925641853622822556, 3.336384251411453099649166941738, 4.02386553490147944708256857489, 5.249170864488892765054651104338, 6.27245705827336029682623909276, 7.075748652336556188175708209017, 7.64837667558758820019628903789, 8.35860654323607559191237462431, 9.40618023664917531645495115003, 9.995504025784742959042195142932, 10.846253072045975029037228935098, 11.71449115264989919961047471419, 12.554201677457667086099322831415, 13.38212143811862677497142322995, 14.039811174784723611654465291610, 14.650752560708404736318190285753, 15.30003882130164893334231840329, 16.01786404230876173570784729282, 17.08638741140471718139584321911, 17.95309248344370784083183284907, 18.61997533590870579373916691337, 19.16421244449982154767607519452, 19.63140231667836073281940310140