L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (0.978 − 0.207i)7-s − 8-s + (−0.669 − 0.743i)13-s + (0.669 + 0.743i)14-s + (−0.5 − 0.866i)16-s + (0.809 − 0.587i)17-s + 19-s + (−0.913 + 0.406i)23-s + (0.309 − 0.951i)26-s + (−0.309 + 0.951i)28-s + (−0.5 − 0.866i)29-s + (−0.978 − 0.207i)31-s + (0.5 − 0.866i)32-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (0.978 − 0.207i)7-s − 8-s + (−0.669 − 0.743i)13-s + (0.669 + 0.743i)14-s + (−0.5 − 0.866i)16-s + (0.809 − 0.587i)17-s + 19-s + (−0.913 + 0.406i)23-s + (0.309 − 0.951i)26-s + (−0.309 + 0.951i)28-s + (−0.5 − 0.866i)29-s + (−0.978 − 0.207i)31-s + (0.5 − 0.866i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0310i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0310i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.889277103 + 0.02931908391i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.889277103 + 0.02931908391i\) |
\(L(1)\) |
\(\approx\) |
\(1.260069259 + 0.4172165494i\) |
\(L(1)\) |
\(\approx\) |
\(1.260069259 + 0.4172165494i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (0.978 - 0.207i)T \) |
| 13 | \( 1 + (-0.669 - 0.743i)T \) |
| 17 | \( 1 + (0.809 - 0.587i)T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (-0.913 + 0.406i)T \) |
| 29 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (-0.978 - 0.207i)T \) |
| 37 | \( 1 + (-0.309 - 0.951i)T \) |
| 41 | \( 1 + (0.913 - 0.406i)T \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
| 47 | \( 1 + (-0.913 - 0.406i)T \) |
| 53 | \( 1 + (0.809 + 0.587i)T \) |
| 59 | \( 1 + (-0.978 - 0.207i)T \) |
| 61 | \( 1 + (0.669 - 0.743i)T \) |
| 67 | \( 1 + (-0.669 - 0.743i)T \) |
| 71 | \( 1 + (0.309 - 0.951i)T \) |
| 73 | \( 1 + (0.809 - 0.587i)T \) |
| 79 | \( 1 + (-0.104 + 0.994i)T \) |
| 83 | \( 1 + (-0.913 - 0.406i)T \) |
| 89 | \( 1 + (-0.809 - 0.587i)T \) |
| 97 | \( 1 + (0.104 - 0.994i)T \) |
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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
−19.64420634028318772145716766400, −18.74785640035956769958572790392, −18.27706369607905145647990971451, −17.553685653752972570484653842887, −16.65671931364253545737423019072, −15.81630170963005174908587069738, −14.67972072939647574964795309009, −14.537228010119163674016645702930, −13.816448614945854142204473223902, −12.86194683626653013845950505862, −12.10914780229990198361579280144, −11.68636226580051855404251802637, −10.89715779957823589985831047452, −10.15211240750910152767038099977, −9.42582422322033866551108047720, −8.64747429006612806627632469695, −7.780882746171230364489005470509, −6.87449265158046596693404910541, −5.71951464370566550618993690855, −5.245167193427050493306788164790, −4.38797426398531935434103829446, −3.67097514347143324981007087489, −2.677186432231974783583024403498, −1.82393938756493091713792188260, −1.17092574974321934778590137114,
0.53405818677682675200681071095, 1.91477261258413095214863037928, 2.97717906805460664002980223835, 3.81515061185841003502180322775, 4.65642634999099151495467115316, 5.454341320509585701733273828800, 5.827582611534612007134484570860, 7.16004124525364153416612939076, 7.6770461662427351570679347830, 8.05763994923008795317110299333, 9.23772619627693905061444715179, 9.79954590674491754466195467057, 10.928178720239215675997208096093, 11.743435071509275237429221743845, 12.33000288699688020674649275534, 13.15401565495325677726973732224, 14.09943435901709024959868840852, 14.31475677588630460639978611815, 15.19228821499211781630094677110, 15.820635060344040255806720369636, 16.63668555634784795150483300285, 17.22327326404665478419104119932, 18.13971251468538790450516450017, 18.21200852316895810864838017437, 19.58502720269603652244766437109