L(s) = 1 | + (−0.258 + 0.965i)2-s + (−0.965 − 0.258i)3-s + (−0.866 − 0.5i)4-s + (0.991 − 0.130i)5-s + (0.5 − 0.866i)6-s + (−0.608 + 0.793i)7-s + (0.707 − 0.707i)8-s + (0.866 + 0.5i)9-s + (−0.130 + 0.991i)10-s + (0.258 + 0.965i)11-s + (0.707 + 0.707i)12-s + (−0.991 + 0.130i)13-s + (−0.608 − 0.793i)14-s + (−0.991 − 0.130i)15-s + (0.5 + 0.866i)16-s + (0.608 + 0.793i)17-s + ⋯ |
L(s) = 1 | + (−0.258 + 0.965i)2-s + (−0.965 − 0.258i)3-s + (−0.866 − 0.5i)4-s + (0.991 − 0.130i)5-s + (0.5 − 0.866i)6-s + (−0.608 + 0.793i)7-s + (0.707 − 0.707i)8-s + (0.866 + 0.5i)9-s + (−0.130 + 0.991i)10-s + (0.258 + 0.965i)11-s + (0.707 + 0.707i)12-s + (−0.991 + 0.130i)13-s + (−0.608 − 0.793i)14-s + (−0.991 − 0.130i)15-s + (0.5 + 0.866i)16-s + (0.608 + 0.793i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.428 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.428 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3260196028 + 0.5152770934i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3260196028 + 0.5152770934i\) |
\(L(1)\) |
\(\approx\) |
\(0.5784224025 + 0.3785660953i\) |
\(L(1)\) |
\(\approx\) |
\(0.5784224025 + 0.3785660953i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 97 | \( 1 \) |
good | 2 | \( 1 + (-0.258 + 0.965i)T \) |
| 3 | \( 1 + (-0.965 - 0.258i)T \) |
| 5 | \( 1 + (0.991 - 0.130i)T \) |
| 7 | \( 1 + (-0.608 + 0.793i)T \) |
| 11 | \( 1 + (0.258 + 0.965i)T \) |
| 13 | \( 1 + (-0.991 + 0.130i)T \) |
| 17 | \( 1 + (0.608 + 0.793i)T \) |
| 19 | \( 1 + (-0.382 + 0.923i)T \) |
| 23 | \( 1 + (-0.793 + 0.608i)T \) |
| 29 | \( 1 + (-0.130 - 0.991i)T \) |
| 31 | \( 1 + (0.965 + 0.258i)T \) |
| 37 | \( 1 + (0.793 + 0.608i)T \) |
| 41 | \( 1 + (0.130 + 0.991i)T \) |
| 43 | \( 1 + (0.866 - 0.5i)T \) |
| 47 | \( 1 + iT \) |
| 53 | \( 1 + (0.258 - 0.965i)T \) |
| 59 | \( 1 + (-0.793 - 0.608i)T \) |
| 61 | \( 1 + (-0.5 - 0.866i)T \) |
| 67 | \( 1 + (0.382 - 0.923i)T \) |
| 71 | \( 1 + (0.130 - 0.991i)T \) |
| 73 | \( 1 + (-0.866 + 0.5i)T \) |
| 79 | \( 1 + (-0.707 + 0.707i)T \) |
| 83 | \( 1 + (-0.608 - 0.793i)T \) |
| 89 | \( 1 + (0.707 - 0.707i)T \) |
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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
−29.597083841906887331656569568604, −29.005032966576110049373233691594, −27.87945993838658280175204825670, −26.83512397507226316501323913221, −26.0669957509785809303347050274, −24.41679909908162343645369474693, −23.06796591866266992074952366453, −22.13567443686250234243762387359, −21.56060096284452648669525470341, −20.35865428174402450259840912475, −19.11476901284578537694483053479, −17.994742723353564095880898953491, −17.07668603075826482198191921154, −16.37807090458784632541354790107, −14.23224903566671241536215950479, −13.21934625358975405714739228507, −12.137938673390055030746754644168, −10.85462804834119853069142237669, −10.10496533123669920070759025065, −9.18922895993196199079189077199, −7.13695601739221471669249543481, −5.711785535148808007890837047633, −4.3778734822740506626845673739, −2.793810380948604369346714285195, −0.81598464559817773464736156998,
1.800556510017643046127329854729, 4.610210870694716803684423449258, 5.82536336296575339377616570849, 6.4260280818012673350649825092, 7.850151071807481341928628523240, 9.631544447719641632583443926137, 10.06570859914647751523203577890, 12.18921185825655768463443034566, 12.9716528785021434255952993080, 14.40616784674800056636137552545, 15.56438995059966594834909045179, 16.82940516450530550143244470545, 17.38335093688456758714442086054, 18.39298632314140838716096381136, 19.37222707000765663681812701067, 21.44810473896669137738260122943, 22.28923305758999429797174858503, 23.12727786879181044106120598853, 24.39788389322992051593739908313, 25.14320180343274881587389329421, 25.968185713295160847212605524026, 27.4719124411501786431301343510, 28.34847457690223201113954461319, 29.03629016294658287606710659250, 30.291365234525330660022813231968