L(s) = 1 | + 5-s − 7-s + 13-s + 17-s + 19-s − 23-s + 25-s − 29-s + 31-s − 35-s − 37-s + 41-s + 43-s − 47-s + 49-s + 53-s + 59-s + 61-s + 65-s − 67-s − 71-s − 73-s − 79-s − 83-s + 85-s − 89-s − 91-s + ⋯ |
L(s) = 1 | + 5-s − 7-s + 13-s + 17-s + 19-s − 23-s + 25-s − 29-s + 31-s − 35-s − 37-s + 41-s + 43-s − 47-s + 49-s + 53-s + 59-s + 61-s + 65-s − 67-s − 71-s − 73-s − 79-s − 83-s + 85-s − 89-s − 91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.389934014\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.389934014\) |
\(L(1)\) |
\(\approx\) |
\(1.198289577\) |
\(L(1)\) |
\(\approx\) |
\(1.198289577\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 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
−25.92648558390805683872667578406, −25.06097349074106557162678051770, −24.122326946002510738419923827736, −22.85755538151709044576609196434, −22.33416689664896284937852727461, −21.18444000985899196111903588413, −20.52679491550270771222564740174, −19.31265843707793477589885707398, −18.43477375556445174306726843021, −17.5873480571489122996940126966, −16.44734217366029771703912066122, −15.85185293812834005886056124938, −14.41533763561748971102289945787, −13.606070877276293607517019559947, −12.82737654800779313730077672825, −11.7035032142437087489452117891, −10.325478105545099956581937529460, −9.71391550926765750108355507425, −8.704704090413726301686463520259, −7.317228757111682871918791963493, −6.13212483464472689841152955043, −5.52676613743764408283123516050, −3.82663000840090336194294558388, −2.753358887465646887815564203757, −1.28386484190770504592501028245,
1.28386484190770504592501028245, 2.753358887465646887815564203757, 3.82663000840090336194294558388, 5.52676613743764408283123516050, 6.13212483464472689841152955043, 7.317228757111682871918791963493, 8.704704090413726301686463520259, 9.71391550926765750108355507425, 10.325478105545099956581937529460, 11.7035032142437087489452117891, 12.82737654800779313730077672825, 13.606070877276293607517019559947, 14.41533763561748971102289945787, 15.85185293812834005886056124938, 16.44734217366029771703912066122, 17.5873480571489122996940126966, 18.43477375556445174306726843021, 19.31265843707793477589885707398, 20.52679491550270771222564740174, 21.18444000985899196111903588413, 22.33416689664896284937852727461, 22.85755538151709044576609196434, 24.122326946002510738419923827736, 25.06097349074106557162678051770, 25.92648558390805683872667578406