| L(s) = 1 | + 5.27·2-s + 19.8·4-s + 10.5·5-s + 62.3·8-s + 55.6·10-s − 34.7·11-s + 37.2·13-s + 170.·16-s − 10.5·17-s + 58.5·19-s + 209.·20-s − 183.·22-s + 125.·23-s − 13.7·25-s + 196.·26-s + 35.4·29-s − 291.·31-s + 399.·32-s − 55.6·34-s − 259.·37-s + 309.·38-s + 658.·40-s − 338.·41-s + 6.80·43-s − 688.·44-s + 661.·46-s + 250.·47-s + ⋯ |
| L(s) = 1 | + 1.86·2-s + 2.47·4-s + 0.943·5-s + 2.75·8-s + 1.75·10-s − 0.952·11-s + 0.795·13-s + 2.66·16-s − 0.150·17-s + 0.707·19-s + 2.33·20-s − 1.77·22-s + 1.13·23-s − 0.109·25-s + 1.48·26-s + 0.226·29-s − 1.69·31-s + 2.20·32-s − 0.280·34-s − 1.15·37-s + 1.31·38-s + 2.60·40-s − 1.28·41-s + 0.0241·43-s − 2.36·44-s + 2.11·46-s + 0.778·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.225207247\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.225207247\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 5.27T + 8T^{2} \) |
| 5 | \( 1 - 10.5T + 125T^{2} \) |
| 11 | \( 1 + 34.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 37.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 10.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 58.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 125.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 35.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 291.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 259.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 338.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 6.80T + 7.95e4T^{2} \) |
| 47 | \( 1 - 250.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 536.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 35.8T + 2.05e5T^{2} \) |
| 61 | \( 1 + 57.7T + 2.26e5T^{2} \) |
| 67 | \( 1 - 481.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 363.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 581.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 693.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.33e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 353.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.44e3T + 9.12e5T^{2} \) |
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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
−10.90583753399711327422700441534, −10.24971021793301382508582500496, −8.900585609738174508222466442371, −7.46977265758075947592952363114, −6.62120739731328617301461187962, −5.53639256926580118373420190407, −5.20872362483547187682229648869, −3.80196629205402709802203622507, −2.80268322442694725187351413560, −1.68227702885123946317957318959,
1.68227702885123946317957318959, 2.80268322442694725187351413560, 3.80196629205402709802203622507, 5.20872362483547187682229648869, 5.53639256926580118373420190407, 6.62120739731328617301461187962, 7.46977265758075947592952363114, 8.900585609738174508222466442371, 10.24971021793301382508582500496, 10.90583753399711327422700441534
