| L(s) = 1 | + 3-s − 5-s + 7-s + 9-s − 11-s + 13-s − 15-s + 17-s + 19-s + 21-s − 23-s + 25-s + 27-s + 29-s − 31-s − 33-s − 35-s − 37-s + 39-s − 41-s + 43-s − 45-s + 47-s + 49-s + 51-s − 53-s + 55-s + ⋯ |
| L(s) = 1 | + 3-s − 5-s + 7-s + 9-s − 11-s + 13-s − 15-s + 17-s + 19-s + 21-s − 23-s + 25-s + 27-s + 29-s − 31-s − 33-s − 35-s − 37-s + 39-s − 41-s + 43-s − 45-s + 47-s + 49-s + 51-s − 53-s + 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.347715141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.347715141\) |
| \(L(1)\) |
\(\approx\) |
\(1.567737324\) |
| \(L(1)\) |
\(\approx\) |
\(1.567737324\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 257 | \( 1 \) |
| good | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 - T \) |
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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
−21.08589847395075652940214026368, −20.545569081349340493684118470, −20.05634619008183950426083179029, −18.93965310625831757587644453749, −18.49196488912158102041665042414, −17.77205831030879270816753029305, −16.32862011675323933469391840294, −15.77508865305808912383647143430, −15.17742277942305912480074113802, −14.16493882390796324011025469594, −13.795060217587440380038378161915, −12.59371081771754980663178934178, −11.93588996160791097271917656661, −10.909702218057130272792583365553, −10.24646771204267018543448875949, −9.06399432130143389016521472904, −8.16239779757466145772621759602, −7.88071122242052974363995281659, −7.075898329622363359653248521930, −5.58638327789774847016088703485, −4.67770499166111090887078614396, −3.72762715517698166950750437480, −3.06395429971113927845755569670, −1.85113509417261511167472766342, −0.84218598501771881733757014515,
0.84218598501771881733757014515, 1.85113509417261511167472766342, 3.06395429971113927845755569670, 3.72762715517698166950750437480, 4.67770499166111090887078614396, 5.58638327789774847016088703485, 7.075898329622363359653248521930, 7.88071122242052974363995281659, 8.16239779757466145772621759602, 9.06399432130143389016521472904, 10.24646771204267018543448875949, 10.909702218057130272792583365553, 11.93588996160791097271917656661, 12.59371081771754980663178934178, 13.795060217587440380038378161915, 14.16493882390796324011025469594, 15.17742277942305912480074113802, 15.77508865305808912383647143430, 16.32862011675323933469391840294, 17.77205831030879270816753029305, 18.49196488912158102041665042414, 18.93965310625831757587644453749, 20.05634619008183950426083179029, 20.545569081349340493684118470, 21.08589847395075652940214026368
