L(s) = 1 | − 3-s + 5-s + 7-s + 9-s − 13-s − 15-s − 17-s + 19-s − 21-s − 23-s + 25-s − 27-s − 29-s − 31-s + 35-s + 37-s + 39-s − 41-s + 43-s + 45-s − 47-s + 49-s + 51-s + 53-s − 57-s − 59-s − 61-s + ⋯ |
L(s) = 1 | − 3-s + 5-s + 7-s + 9-s − 13-s − 15-s − 17-s + 19-s − 21-s − 23-s + 25-s − 27-s − 29-s − 31-s + 35-s + 37-s + 39-s − 41-s + 43-s + 45-s − 47-s + 49-s + 51-s + 53-s − 57-s − 59-s − 61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7600622838\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7600622838\) |
\(L(1)\) |
\(\approx\) |
\(0.9024906449\) |
\(L(1)\) |
\(\approx\) |
\(0.9024906449\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
good | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 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
−34.15346729684272106043380578223, −33.60625213311540800317874948376, −32.488723456435195652185644207532, −30.7837343166950765025407660372, −29.60681027223561542267337147912, −28.77893942445047647072544291977, −27.60237248544315232420294111024, −26.442423451163784130338143627684, −24.681483218497409008454757604732, −24.04471027020575246774989117385, −22.32883476081719799605147614760, −21.66563972555785204780870557439, −20.30638125042707154146627647832, −18.269033097875990316507319931584, −17.61972287425101875532373646025, −16.51922470349364623489658320724, −14.83198533873399906847331308472, −13.41643453051203828157973988185, −11.956373727740704105680429338707, −10.74074979750747559997644955886, −9.44566152389131881369715831192, −7.38790752248483336704332902315, −5.81981644686087148920681986787, −4.70920812840911269661655082592, −1.869939273182751683408212296752,
1.869939273182751683408212296752, 4.70920812840911269661655082592, 5.81981644686087148920681986787, 7.38790752248483336704332902315, 9.44566152389131881369715831192, 10.74074979750747559997644955886, 11.956373727740704105680429338707, 13.41643453051203828157973988185, 14.83198533873399906847331308472, 16.51922470349364623489658320724, 17.61972287425101875532373646025, 18.269033097875990316507319931584, 20.30638125042707154146627647832, 21.66563972555785204780870557439, 22.32883476081719799605147614760, 24.04471027020575246774989117385, 24.681483218497409008454757604732, 26.442423451163784130338143627684, 27.60237248544315232420294111024, 28.77893942445047647072544291977, 29.60681027223561542267337147912, 30.7837343166950765025407660372, 32.488723456435195652185644207532, 33.60625213311540800317874948376, 34.15346729684272106043380578223