| L(s) = 1 | − 4.30·2-s + 10.5·4-s + 16.7·5-s + 7·7-s − 10.8·8-s − 71.9·10-s + 29.7·11-s + 74.3·13-s − 30.1·14-s − 37.4·16-s − 125.·17-s + 111.·19-s + 175.·20-s − 127.·22-s − 23·23-s + 154.·25-s − 320.·26-s + 73.6·28-s − 121.·29-s + 113.·31-s + 248.·32-s + 538.·34-s + 117.·35-s + 221.·37-s − 480.·38-s − 181.·40-s + 203.·41-s + ⋯ |
| L(s) = 1 | − 1.52·2-s + 1.31·4-s + 1.49·5-s + 0.377·7-s − 0.479·8-s − 2.27·10-s + 0.815·11-s + 1.58·13-s − 0.575·14-s − 0.585·16-s − 1.78·17-s + 1.34·19-s + 1.96·20-s − 1.24·22-s − 0.208·23-s + 1.23·25-s − 2.41·26-s + 0.497·28-s − 0.775·29-s + 0.656·31-s + 1.37·32-s + 2.71·34-s + 0.565·35-s + 0.982·37-s − 2.05·38-s − 0.716·40-s + 0.775·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.733182436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.733182436\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 23 | \( 1 + 23T \) |
| good | 2 | \( 1 + 4.30T + 8T^{2} \) |
| 5 | \( 1 - 16.7T + 125T^{2} \) |
| 11 | \( 1 - 29.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 74.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 125.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 111.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 121.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 113.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 221.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 203.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 360.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 391.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 317.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 329.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 631.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 153.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 767.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.02e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.14e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 225.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 437.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 709.T + 9.12e5T^{2} \) |
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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
−9.387655829908081344325675203843, −8.603371793704656604109448518412, −7.82015874100210414893242841605, −6.62843718656652187207277640166, −6.31596783675719211118567596219, −5.20979700747039819590843601724, −3.96732838818431803949145363767, −2.43782234483993891499423476272, −1.61207224061337057713487017455, −0.897935268140589877221583634549,
0.897935268140589877221583634549, 1.61207224061337057713487017455, 2.43782234483993891499423476272, 3.96732838818431803949145363767, 5.20979700747039819590843601724, 6.31596783675719211118567596219, 6.62843718656652187207277640166, 7.82015874100210414893242841605, 8.603371793704656604109448518412, 9.387655829908081344325675203843
