| L(s) = 1 | + 2.64·3-s + 4.08·5-s − 3.94·7-s + 4.01·9-s + 3.83·11-s − 1.75·13-s + 10.8·15-s + 0.590·19-s − 10.4·21-s + 3.92·23-s + 11.7·25-s + 2.67·27-s − 8.02·29-s + 5.59·31-s + 10.1·33-s − 16.1·35-s + 6.07·37-s − 4.64·39-s + 5.95·41-s + 4.96·43-s + 16.3·45-s − 0.492·47-s + 8.53·49-s − 6.70·53-s + 15.6·55-s + 1.56·57-s + 2.72·59-s + ⋯ |
| L(s) = 1 | + 1.52·3-s + 1.82·5-s − 1.48·7-s + 1.33·9-s + 1.15·11-s − 0.486·13-s + 2.79·15-s + 0.135·19-s − 2.27·21-s + 0.817·23-s + 2.34·25-s + 0.514·27-s − 1.48·29-s + 1.00·31-s + 1.76·33-s − 2.72·35-s + 0.999·37-s − 0.744·39-s + 0.930·41-s + 0.757·43-s + 2.44·45-s − 0.0718·47-s + 1.21·49-s − 0.920·53-s + 2.11·55-s + 0.207·57-s + 0.355·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.153119551\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.153119551\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 2.64T + 3T^{2} \) |
| 5 | \( 1 - 4.08T + 5T^{2} \) |
| 7 | \( 1 + 3.94T + 7T^{2} \) |
| 11 | \( 1 - 3.83T + 11T^{2} \) |
| 13 | \( 1 + 1.75T + 13T^{2} \) |
| 19 | \( 1 - 0.590T + 19T^{2} \) |
| 23 | \( 1 - 3.92T + 23T^{2} \) |
| 29 | \( 1 + 8.02T + 29T^{2} \) |
| 31 | \( 1 - 5.59T + 31T^{2} \) |
| 37 | \( 1 - 6.07T + 37T^{2} \) |
| 41 | \( 1 - 5.95T + 41T^{2} \) |
| 43 | \( 1 - 4.96T + 43T^{2} \) |
| 47 | \( 1 + 0.492T + 47T^{2} \) |
| 53 | \( 1 + 6.70T + 53T^{2} \) |
| 59 | \( 1 - 2.72T + 59T^{2} \) |
| 61 | \( 1 + 6.67T + 61T^{2} \) |
| 67 | \( 1 - 0.855T + 67T^{2} \) |
| 71 | \( 1 + 0.354T + 71T^{2} \) |
| 73 | \( 1 + 2.28T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 - 4.26T + 83T^{2} \) |
| 89 | \( 1 + 1.77T + 89T^{2} \) |
| 97 | \( 1 - 3.42T + 97T^{2} \) |
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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
−7.64043074517587111575326444547, −7.04133342096220967779465730961, −6.25301468867861553254179131853, −6.00770353213200839487205973747, −4.91774776398066954572386053479, −3.95472643448962048968236873273, −3.19525051816151731389447605271, −2.64375549030295280296803145627, −2.00876113756404520875665167001, −1.05585256452295787704717689366,
1.05585256452295787704717689366, 2.00876113756404520875665167001, 2.64375549030295280296803145627, 3.19525051816151731389447605271, 3.95472643448962048968236873273, 4.91774776398066954572386053479, 6.00770353213200839487205973747, 6.25301468867861553254179131853, 7.04133342096220967779465730961, 7.64043074517587111575326444547
