| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 9-s + 10-s − 11-s − 12-s + 13-s − 14-s − 15-s + 16-s − 17-s + 18-s + 19-s + 20-s + 21-s − 22-s + 23-s − 24-s + 25-s + 26-s − 27-s − 28-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 9-s + 10-s − 11-s − 12-s + 13-s − 14-s − 15-s + 16-s − 17-s + 18-s + 19-s + 20-s + 21-s − 22-s + 23-s − 24-s + 25-s + 26-s − 27-s − 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 751 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 751 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.475651110\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.475651110\) |
| \(L(1)\) |
\(\approx\) |
\(1.719575161\) |
| \(L(1)\) |
\(\approx\) |
\(1.719575161\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 751 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 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
−22.28198916841830033894902371589, −21.72274900066539295365149965366, −20.84349506119742704795718745210, −20.21868953806723586839653617323, −18.82059749824938819310811605770, −18.21821085583733691375949081161, −17.199099861529140717339861911603, −16.33722333858207747833601109482, −15.865707116919839568370487641564, −14.982621439531572525994905343, −13.58563792787864190265431399208, −13.17424204114809419556693821111, −12.702288766198074036947952773173, −11.447364101253125436108840223483, −10.7814023814473117339267544851, −10.06014507030296056953193894296, −9.041192787617262615474630069496, −7.32741203910052758801284487977, −6.70063905430819696672796331255, −5.69633293243583257957336622384, −5.47350152729541018410749015929, −4.23306401456501442707456162858, −3.1292280546105368039070888522, −2.08220168742220260687282574652, −0.83929899312793750932141322795,
0.83929899312793750932141322795, 2.08220168742220260687282574652, 3.1292280546105368039070888522, 4.23306401456501442707456162858, 5.47350152729541018410749015929, 5.69633293243583257957336622384, 6.70063905430819696672796331255, 7.32741203910052758801284487977, 9.041192787617262615474630069496, 10.06014507030296056953193894296, 10.7814023814473117339267544851, 11.447364101253125436108840223483, 12.702288766198074036947952773173, 13.17424204114809419556693821111, 13.58563792787864190265431399208, 14.982621439531572525994905343, 15.865707116919839568370487641564, 16.33722333858207747833601109482, 17.199099861529140717339861911603, 18.21821085583733691375949081161, 18.82059749824938819310811605770, 20.21868953806723586839653617323, 20.84349506119742704795718745210, 21.72274900066539295365149965366, 22.28198916841830033894902371589
