L(s) = 1 | − 3-s − 5-s − 7-s + 9-s − 13-s + 15-s + 17-s − 19-s + 21-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 − 63-s + ⋯ |
L(s) = 1 | − 3-s − 5-s − 7-s + 9-s − 13-s + 15-s + 17-s − 19-s + 21-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 − 63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.04870024580\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.04870024580\) |
\(L(1)\) |
\(\approx\) |
\(0.3950204758\) |
\(L(1)\) |
\(\approx\) |
\(0.3950204758\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 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
−21.736114788312010332020366442357, −20.622925934335468248105636703416, −19.7055223823745399107469714046, −18.97138417564053962930515190614, −18.555246096564749490587386570375, −17.24119391098984576249949940021, −16.719697087666148023882231624847, −16.0906429187151965972423479002, −15.260432942255408030334558656580, −14.56068176517078421782452581459, −13.109702240349699896297774945023, −12.54450584739198238283459817912, −11.94513279184025215970281069887, −11.10193587325115487102688849534, −10.230672534201348004051587683609, −9.56360743843928311920397302480, −8.35746741597925856612341031233, −7.28463458774917937101622600666, −6.826335033312680050928120810603, −5.747402044508024193530845353282, −4.90461665078782728657915435459, −3.928745557801679677069526183274, −3.13145005148115808349175463316, −1.63007179383371076139374007318, −0.10666881536242395641005015805,
0.10666881536242395641005015805, 1.63007179383371076139374007318, 3.13145005148115808349175463316, 3.928745557801679677069526183274, 4.90461665078782728657915435459, 5.747402044508024193530845353282, 6.826335033312680050928120810603, 7.28463458774917937101622600666, 8.35746741597925856612341031233, 9.56360743843928311920397302480, 10.230672534201348004051587683609, 11.10193587325115487102688849534, 11.94513279184025215970281069887, 12.54450584739198238283459817912, 13.109702240349699896297774945023, 14.56068176517078421782452581459, 15.260432942255408030334558656580, 16.0906429187151965972423479002, 16.719697087666148023882231624847, 17.24119391098984576249949940021, 18.555246096564749490587386570375, 18.97138417564053962930515190614, 19.7055223823745399107469714046, 20.622925934335468248105636703416, 21.736114788312010332020366442357