| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s − 11-s + 12-s + 15-s + 16-s − 17-s + 18-s + 20-s − 22-s + 23-s + 24-s + 25-s + 27-s − 29-s + 30-s − 31-s + 32-s − 33-s − 34-s + 36-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s − 11-s + 12-s + 15-s + 16-s − 17-s + 18-s + 20-s − 22-s + 23-s + 24-s + 25-s + 27-s − 29-s + 30-s − 31-s + 32-s − 33-s − 34-s + 36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(5.485046587\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.485046587\) |
| \(L(1)\) |
\(\approx\) |
\(3.090063213\) |
| \(L(1)\) |
\(\approx\) |
\(3.090063213\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 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
−20.463612661263978907704056576377, −19.92708293996445195206570134727, −18.833661428598869609663873021438, −18.29170524774732556359977431034, −17.20010737892209897759824022113, −16.415728550716005292623082645683, −15.4453053283689567922948663663, −15.0782729153791899086439983451, −14.146092222662917403246840842713, −13.63483608688376387808514230613, −12.90738643101342180008355385368, −12.638193191374443672452967858271, −11.08789413217670756711781285377, −10.658257163138566991137072007052, −9.6401291918048351200189727766, −8.97617648095196207361234760045, −7.89916603737881826587931772833, −7.17065150935229486654552095309, −6.37144458188859794805335182221, −5.40686510869533426538657792639, −4.73492876088125932652211004441, −3.755563159107694811775706509915, −2.77891299642946434085222088506, −2.29229340648354120943418259824, −1.4206010613486802709621688509,
1.4206010613486802709621688509, 2.29229340648354120943418259824, 2.77891299642946434085222088506, 3.755563159107694811775706509915, 4.73492876088125932652211004441, 5.40686510869533426538657792639, 6.37144458188859794805335182221, 7.17065150935229486654552095309, 7.89916603737881826587931772833, 8.97617648095196207361234760045, 9.6401291918048351200189727766, 10.658257163138566991137072007052, 11.08789413217670756711781285377, 12.638193191374443672452967858271, 12.90738643101342180008355385368, 13.63483608688376387808514230613, 14.146092222662917403246840842713, 15.0782729153791899086439983451, 15.4453053283689567922948663663, 16.415728550716005292623082645683, 17.20010737892209897759824022113, 18.29170524774732556359977431034, 18.833661428598869609663873021438, 19.92708293996445195206570134727, 20.463612661263978907704056576377
