L(s) = 1 | + 2.56·3-s + 5-s + 2.74·7-s + 3.56·9-s + 3.38·11-s − 2.46·13-s + 2.56·15-s − 2.64·17-s − 3.38·19-s + 7.02·21-s + 23-s + 25-s + 1.43·27-s + 9.20·29-s + 5.10·31-s + 8.67·33-s + 2.74·35-s + 5.50·37-s − 6.31·39-s − 1.20·41-s − 3.02·43-s + 3.56·45-s − 8.21·47-s + 0.521·49-s − 6.77·51-s − 10.5·53-s + 3.38·55-s + ⋯ |
L(s) = 1 | + 1.47·3-s + 0.447·5-s + 1.03·7-s + 1.18·9-s + 1.02·11-s − 0.683·13-s + 0.661·15-s − 0.641·17-s − 0.777·19-s + 1.53·21-s + 0.208·23-s + 0.200·25-s + 0.276·27-s + 1.70·29-s + 0.917·31-s + 1.51·33-s + 0.463·35-s + 0.904·37-s − 1.01·39-s − 0.188·41-s − 0.461·43-s + 0.530·45-s − 1.19·47-s + 0.0744·49-s − 0.948·51-s − 1.44·53-s + 0.456·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.624704339\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.624704339\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 - 2.56T + 3T^{2} \) |
| 7 | \( 1 - 2.74T + 7T^{2} \) |
| 11 | \( 1 - 3.38T + 11T^{2} \) |
| 13 | \( 1 + 2.46T + 13T^{2} \) |
| 17 | \( 1 + 2.64T + 17T^{2} \) |
| 19 | \( 1 + 3.38T + 19T^{2} \) |
| 29 | \( 1 - 9.20T + 29T^{2} \) |
| 31 | \( 1 - 5.10T + 31T^{2} \) |
| 37 | \( 1 - 5.50T + 37T^{2} \) |
| 41 | \( 1 + 1.20T + 41T^{2} \) |
| 43 | \( 1 + 3.02T + 43T^{2} \) |
| 47 | \( 1 + 8.21T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 + 8.28T + 59T^{2} \) |
| 61 | \( 1 - 0.263T + 61T^{2} \) |
| 67 | \( 1 - 7.66T + 67T^{2} \) |
| 71 | \( 1 - 0.0150T + 71T^{2} \) |
| 73 | \( 1 + 5.53T + 73T^{2} \) |
| 79 | \( 1 - 8.67T + 79T^{2} \) |
| 83 | \( 1 - 3.52T + 83T^{2} \) |
| 89 | \( 1 + 4.66T + 89T^{2} \) |
| 97 | \( 1 + 4.09T + 97T^{2} \) |
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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
−9.233064567883034173362817423375, −8.240498085989109290796165453304, −8.165680096500988705245310489506, −6.93257665704814491587295575775, −6.28875097909043620109947373618, −4.82008406092403959906473805082, −4.35255270672077822236042807390, −3.13763270738465019851342535570, −2.28603528089410236322979882905, −1.43634937329664052385117847947,
1.43634937329664052385117847947, 2.28603528089410236322979882905, 3.13763270738465019851342535570, 4.35255270672077822236042807390, 4.82008406092403959906473805082, 6.28875097909043620109947373618, 6.93257665704814491587295575775, 8.165680096500988705245310489506, 8.240498085989109290796165453304, 9.233064567883034173362817423375