| L(s) = 1 | − 5.67·5-s − 22.9·7-s − 32.5·11-s − 13·13-s − 107.·17-s − 116.·19-s − 65.1·23-s − 92.7·25-s − 92.5·29-s − 51.1·31-s + 130.·35-s + 267.·37-s + 392.·41-s − 317.·43-s − 114.·47-s + 184.·49-s + 618·53-s + 184.·55-s − 600.·59-s − 857.·61-s + 73.7·65-s − 422.·67-s − 428.·71-s − 220.·73-s + 748.·77-s + 235.·79-s + 64.3·83-s + ⋯ |
| L(s) = 1 | − 0.507·5-s − 1.24·7-s − 0.893·11-s − 0.277·13-s − 1.53·17-s − 1.41·19-s − 0.591·23-s − 0.742·25-s − 0.592·29-s − 0.296·31-s + 0.629·35-s + 1.18·37-s + 1.49·41-s − 1.12·43-s − 0.356·47-s + 0.538·49-s + 1.60·53-s + 0.453·55-s − 1.32·59-s − 1.79·61-s + 0.140·65-s − 0.769·67-s − 0.716·71-s − 0.353·73-s + 1.10·77-s + 0.335·79-s + 0.0851·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1196544793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1196544793\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 5 | \( 1 + 5.67T + 125T^{2} \) |
| 7 | \( 1 + 22.9T + 343T^{2} \) |
| 11 | \( 1 + 32.5T + 1.33e3T^{2} \) |
| 17 | \( 1 + 107.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 116.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 65.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 92.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 51.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 267.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 392.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 317.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 114.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 618T + 1.48e5T^{2} \) |
| 59 | \( 1 + 600.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 857.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 422.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 428.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 220.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 235.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 64.3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.07e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 703.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.942155868486544380809351552160, −8.034909988727955804996373449157, −7.33758244260778956653882935227, −6.41375570647146579173167447691, −5.88973269720466161691608412312, −4.58896235578791201598832811721, −3.97143395146052740524719131118, −2.87289498792147773975042770647, −2.06898921329447636407730059117, −0.14842948319467762912087656992,
0.14842948319467762912087656992, 2.06898921329447636407730059117, 2.87289498792147773975042770647, 3.97143395146052740524719131118, 4.58896235578791201598832811721, 5.88973269720466161691608412312, 6.41375570647146579173167447691, 7.33758244260778956653882935227, 8.034909988727955804996373449157, 8.942155868486544380809351552160
