| L(s) = 1 | + 3-s − 5-s + 9-s + 13-s − 15-s + 17-s − 19-s − 23-s + 25-s + 27-s − 29-s + 31-s + 37-s + 39-s + 41-s + 43-s − 45-s + 47-s + 51-s + 53-s − 57-s + 59-s + 61-s − 65-s − 67-s − 69-s − 71-s + ⋯ |
| L(s) = 1 | + 3-s − 5-s + 9-s + 13-s − 15-s + 17-s − 19-s − 23-s + 25-s + 27-s − 29-s + 31-s + 37-s + 39-s + 41-s + 43-s − 45-s + 47-s + 51-s + 53-s − 57-s + 59-s + 61-s − 65-s − 67-s − 69-s − 71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.555298498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.555298498\) |
| \(L(1)\) |
\(\approx\) |
\(1.432070839\) |
| \(L(1)\) |
\(\approx\) |
\(1.432070839\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 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 | \( 1 - T \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−25.21198335573161175361898524985, −24.074447592719771406762460734919, −23.47212867335618317388655842414, −22.42817290031799162243952757337, −21.15353089265036580753912303506, −20.58847603695695065755022285610, −19.56861081468061813385160139175, −18.97307765003057045225718912708, −18.126923169470896330123668642598, −16.60049369325141955072613386934, −15.808072740995980947095631203611, −14.99164459954427941213072280671, −14.17362696708805234664373884652, −13.10089421756687673200913667634, −12.2254766815761331723357362716, −11.08453751233115490906075423862, −10.037034293612516254391383122663, −8.83294079150453506686817394536, −8.094950273856194188874543331687, −7.31061324002411018271832190411, −5.98079546187723518138244861730, −4.28107002051492708582831134163, −3.6900900245573321137089676888, −2.45778014917415654791015162030, −0.95233085156013228844405374138,
0.95233085156013228844405374138, 2.45778014917415654791015162030, 3.6900900245573321137089676888, 4.28107002051492708582831134163, 5.98079546187723518138244861730, 7.31061324002411018271832190411, 8.094950273856194188874543331687, 8.83294079150453506686817394536, 10.037034293612516254391383122663, 11.08453751233115490906075423862, 12.2254766815761331723357362716, 13.10089421756687673200913667634, 14.17362696708805234664373884652, 14.99164459954427941213072280671, 15.808072740995980947095631203611, 16.60049369325141955072613386934, 18.126923169470896330123668642598, 18.97307765003057045225718912708, 19.56861081468061813385160139175, 20.58847603695695065755022285610, 21.15353089265036580753912303506, 22.42817290031799162243952757337, 23.47212867335618317388655842414, 24.074447592719771406762460734919, 25.21198335573161175361898524985
