| L(s) = 1 | − 4.83·3-s − 11.8·5-s − 7·7-s − 3.62·9-s − 38.0·11-s − 68.7·13-s + 57.4·15-s + 10.8·17-s + 22.5·19-s + 33.8·21-s − 17.4·23-s + 16.1·25-s + 148.·27-s − 215.·29-s − 107.·31-s + 184·33-s + 83.1·35-s − 396.·37-s + 332.·39-s + 171.·41-s + 500.·43-s + 43.0·45-s + 300.·47-s + 49·49-s − 52.6·51-s − 35.1·53-s + 452.·55-s + ⋯ |
| L(s) = 1 | − 0.930·3-s − 1.06·5-s − 0.377·7-s − 0.134·9-s − 1.04·11-s − 1.46·13-s + 0.988·15-s + 0.155·17-s + 0.272·19-s + 0.351·21-s − 0.158·23-s + 0.129·25-s + 1.05·27-s − 1.38·29-s − 0.625·31-s + 0.970·33-s + 0.401·35-s − 1.75·37-s + 1.36·39-s + 0.654·41-s + 1.77·43-s + 0.142·45-s + 0.932·47-s + 0.142·49-s − 0.144·51-s − 0.0909·53-s + 1.10·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2805188790\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2805188790\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 3 | \( 1 + 4.83T + 27T^{2} \) |
| 5 | \( 1 + 11.8T + 125T^{2} \) |
| 11 | \( 1 + 38.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 68.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 10.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 22.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 17.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 215.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 107.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 396.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 171.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 500.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 300.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 35.1T + 1.48e5T^{2} \) |
| 59 | \( 1 - 356.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 882.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 444.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 72.5T + 3.57e5T^{2} \) |
| 73 | \( 1 + 649.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 73.0T + 4.93e5T^{2} \) |
| 83 | \( 1 - 142.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 620.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 907.T + 9.12e5T^{2} \) |
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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
−10.81322426449406331457425456098, −10.00368798769030187589836796984, −8.863408850294086318757791621418, −7.63416715499364042498047840574, −7.18926329051617644303867537959, −5.74424762247203823559300396988, −5.08616755580471923085645961237, −3.85932582049894310651415141282, −2.54754618416332929168536839171, −0.32919140653559681334308315004,
0.32919140653559681334308315004, 2.54754618416332929168536839171, 3.85932582049894310651415141282, 5.08616755580471923085645961237, 5.74424762247203823559300396988, 7.18926329051617644303867537959, 7.63416715499364042498047840574, 8.863408850294086318757791621418, 10.00368798769030187589836796984, 10.81322426449406331457425456098
