| L(s) = 1 | − 0.980·3-s − 1.69·5-s + 1.30·7-s − 2.03·9-s + 3.27·11-s + 4.30·13-s + 1.65·15-s − 1.02·17-s + 7.01·19-s − 1.27·21-s − 0.835·23-s − 2.13·25-s + 4.93·27-s − 1.11·29-s + 3.97·31-s − 3.20·33-s − 2.20·35-s − 11.3·37-s − 4.22·39-s + 1.22·41-s − 10.8·43-s + 3.44·45-s + 0.151·47-s − 5.29·49-s + 1.00·51-s + 3.02·53-s − 5.53·55-s + ⋯ |
| L(s) = 1 | − 0.565·3-s − 0.756·5-s + 0.493·7-s − 0.679·9-s + 0.986·11-s + 1.19·13-s + 0.427·15-s − 0.248·17-s + 1.60·19-s − 0.279·21-s − 0.174·23-s − 0.427·25-s + 0.950·27-s − 0.207·29-s + 0.713·31-s − 0.558·33-s − 0.373·35-s − 1.85·37-s − 0.675·39-s + 0.191·41-s − 1.65·43-s + 0.514·45-s + 0.0221·47-s − 0.756·49-s + 0.140·51-s + 0.414·53-s − 0.745·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.387425892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.387425892\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 3 | \( 1 + 0.980T + 3T^{2} \) |
| 5 | \( 1 + 1.69T + 5T^{2} \) |
| 7 | \( 1 - 1.30T + 7T^{2} \) |
| 11 | \( 1 - 3.27T + 11T^{2} \) |
| 13 | \( 1 - 4.30T + 13T^{2} \) |
| 17 | \( 1 + 1.02T + 17T^{2} \) |
| 19 | \( 1 - 7.01T + 19T^{2} \) |
| 23 | \( 1 + 0.835T + 23T^{2} \) |
| 29 | \( 1 + 1.11T + 29T^{2} \) |
| 31 | \( 1 - 3.97T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 - 1.22T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 - 0.151T + 47T^{2} \) |
| 53 | \( 1 - 3.02T + 53T^{2} \) |
| 59 | \( 1 - 4.15T + 59T^{2} \) |
| 61 | \( 1 - 5.62T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 - 1.66T + 79T^{2} \) |
| 83 | \( 1 - 2.34T + 83T^{2} \) |
| 89 | \( 1 - 18.1T + 89T^{2} \) |
| 97 | \( 1 + 7.17T + 97T^{2} \) |
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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
−8.441959813414495126596211941588, −7.85561776251396279670635550337, −6.91053887225551276039509746647, −6.30034918148247125740395786253, −5.46118103872169987599008547425, −4.80981187089694796256452270009, −3.73564800047826273608521022089, −3.29687527374595113750608389191, −1.76339137493284522506843525437, −0.72673181315553704957953486644,
0.72673181315553704957953486644, 1.76339137493284522506843525437, 3.29687527374595113750608389191, 3.73564800047826273608521022089, 4.80981187089694796256452270009, 5.46118103872169987599008547425, 6.30034918148247125740395786253, 6.91053887225551276039509746647, 7.85561776251396279670635550337, 8.441959813414495126596211941588
