L(s) = 1 | + (0.173 − 0.984i)2-s + (0.5 − 0.866i)3-s + (−0.939 − 0.342i)4-s + (−0.866 + 0.5i)5-s + (−0.766 − 0.642i)6-s + (−0.5 − 0.866i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)9-s + (0.342 + 0.939i)10-s + (−0.342 + 0.939i)11-s + (−0.766 + 0.642i)12-s + (−0.5 + 0.866i)13-s + (−0.939 + 0.342i)14-s + i·15-s + (0.766 + 0.642i)16-s + (0.939 − 0.342i)17-s + ⋯ |
L(s) = 1 | + (0.173 − 0.984i)2-s + (0.5 − 0.866i)3-s + (−0.939 − 0.342i)4-s + (−0.866 + 0.5i)5-s + (−0.766 − 0.642i)6-s + (−0.5 − 0.866i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)9-s + (0.342 + 0.939i)10-s + (−0.342 + 0.939i)11-s + (−0.766 + 0.642i)12-s + (−0.5 + 0.866i)13-s + (−0.939 + 0.342i)14-s + i·15-s + (0.766 + 0.642i)16-s + (0.939 − 0.342i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.808 + 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.808 + 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.3103562852 - 0.9540741376i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.3103562852 - 0.9540741376i\) |
\(L(1)\) |
\(\approx\) |
\(0.5962498600 - 0.6910142165i\) |
\(L(1)\) |
\(\approx\) |
\(0.5962498600 - 0.6910142165i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (0.173 - 0.984i)T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + (-0.342 + 0.939i)T \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
| 17 | \( 1 + (0.939 - 0.342i)T \) |
| 19 | \( 1 + (0.766 - 0.642i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.642 - 0.766i)T \) |
| 31 | \( 1 + (-0.342 + 0.939i)T \) |
| 41 | \( 1 + (0.642 + 0.766i)T \) |
| 43 | \( 1 + (-0.866 - 0.5i)T \) |
| 47 | \( 1 + (-0.342 + 0.939i)T \) |
| 53 | \( 1 + (0.984 - 0.173i)T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 - iT \) |
| 67 | \( 1 + (-0.984 - 0.173i)T \) |
| 71 | \( 1 + (-0.173 - 0.984i)T \) |
| 73 | \( 1 + (-0.173 - 0.984i)T \) |
| 79 | \( 1 + (0.173 - 0.984i)T \) |
| 83 | \( 1 + (-0.766 + 0.642i)T \) |
| 89 | \( 1 + (0.642 + 0.766i)T \) |
| 97 | \( 1 + (0.642 - 0.766i)T \) |
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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
−18.8769760305440334381995004570, −18.25318516351396859480368766101, −17.13777479955845853924833000694, −16.544964968191943311628632563612, −15.998907743132259808875587375011, −15.64097418383799517471159107385, −14.84491225211664986865837880090, −14.54016513682072682204275348043, −13.42381572534961478907887294354, −12.961839456677862709853550690225, −12.09957605212041778507275927198, −11.47931372016742751181295399291, −10.276189075338485047330602678050, −9.76683500122636110639883579831, −8.85396260617955872908168118382, −8.50066545068958522098401118799, −7.79707631705693595004520764308, −7.24423166249383118840425817915, −5.78645862979674546047424892283, −5.514789652845737348478933742025, −4.91258899853393102728590673446, −3.75477715292268241305349577883, −3.41622139092226488110883906325, −2.71220264597473490139872704098, −0.95610039667689783755904514145,
0.32733974772040551014184084619, 1.15887053009357579574577987102, 2.17702001649951669269254461925, 2.963269362994158262781429026781, 3.39819288421049246475558909739, 4.37610116566359537208310798270, 4.90810445557136919573770170599, 6.27430376242200132933774528029, 7.04567383831847574538739300822, 7.48302251737614524011700453764, 8.25430962593254034111042044848, 9.19786169123387167493387547996, 9.830697082616023848378009971459, 10.49200301203097588320470963860, 11.37219776223937642727741649869, 12.02052354316333745495550351546, 12.438921886020685010706275237769, 13.21087436520782982383464279950, 13.87523217164004340754847575843, 14.48986062003263717408143741771, 14.95325849012984740612326044644, 15.998608801445598772286779466828, 16.8516492570484818556628748792, 17.74210393589703209305213351996, 18.275966016719038161001765742523