| L(s) = 1 | − 5·5-s − 18·7-s − 34·11-s + 12·13-s + 102·17-s + 164·19-s − 48·23-s + 25·25-s − 146·29-s + 100·31-s + 90·35-s + 328·37-s + 288·41-s + 120·43-s − 16·47-s − 19·49-s + 126·53-s + 170·55-s − 642·59-s + 602·61-s − 60·65-s + 436·67-s − 652·71-s + 1.06e3·73-s + 612·77-s + 388·79-s + 444·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.971·7-s − 0.931·11-s + 0.256·13-s + 1.45·17-s + 1.98·19-s − 0.435·23-s + 1/5·25-s − 0.934·29-s + 0.579·31-s + 0.434·35-s + 1.45·37-s + 1.09·41-s + 0.425·43-s − 0.0496·47-s − 0.0553·49-s + 0.326·53-s + 0.416·55-s − 1.41·59-s + 1.26·61-s − 0.114·65-s + 0.795·67-s − 1.08·71-s + 1.70·73-s + 0.905·77-s + 0.552·79-s + 0.587·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.439147309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.439147309\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 + 18 T + p^{3} T^{2} \) |
| 11 | \( 1 + 34 T + p^{3} T^{2} \) |
| 13 | \( 1 - 12 T + p^{3} T^{2} \) |
| 17 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 19 | \( 1 - 164 T + p^{3} T^{2} \) |
| 23 | \( 1 + 48 T + p^{3} T^{2} \) |
| 29 | \( 1 + 146 T + p^{3} T^{2} \) |
| 31 | \( 1 - 100 T + p^{3} T^{2} \) |
| 37 | \( 1 - 328 T + p^{3} T^{2} \) |
| 41 | \( 1 - 288 T + p^{3} T^{2} \) |
| 43 | \( 1 - 120 T + p^{3} T^{2} \) |
| 47 | \( 1 + 16 T + p^{3} T^{2} \) |
| 53 | \( 1 - 126 T + p^{3} T^{2} \) |
| 59 | \( 1 + 642 T + p^{3} T^{2} \) |
| 61 | \( 1 - 602 T + p^{3} T^{2} \) |
| 67 | \( 1 - 436 T + p^{3} T^{2} \) |
| 71 | \( 1 + 652 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1062 T + p^{3} T^{2} \) |
| 79 | \( 1 - 388 T + p^{3} T^{2} \) |
| 83 | \( 1 - 444 T + p^{3} T^{2} \) |
| 89 | \( 1 - 820 T + p^{3} T^{2} \) |
| 97 | \( 1 + 766 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.03964059877102691472010348394, −9.930749460123633089077607501305, −9.425557487735175138962639193132, −7.944244178915745205506288914739, −7.46021182418504692556773242300, −6.09348363253464390241063558204, −5.21990279329320009327709204193, −3.70596339947156941096040781127, −2.83349890164339911917483728198, −0.803052331988881326898957305575,
0.803052331988881326898957305575, 2.83349890164339911917483728198, 3.70596339947156941096040781127, 5.21990279329320009327709204193, 6.09348363253464390241063558204, 7.46021182418504692556773242300, 7.944244178915745205506288914739, 9.425557487735175138962639193132, 9.930749460123633089077607501305, 11.03964059877102691472010348394
