| L(s) = 1 | − 4.05·2-s + 8.44·4-s − 9.92·5-s − 1.80·8-s + 40.2·10-s − 13.5·11-s + 18.5·13-s − 60.2·16-s − 93.7·17-s + 131.·19-s − 83.7·20-s + 54.9·22-s − 198.·23-s − 26.5·25-s − 75.2·26-s + 188.·29-s − 83.9·31-s + 258.·32-s + 380.·34-s + 80.1·37-s − 534.·38-s + 17.9·40-s − 385.·41-s − 397.·43-s − 114.·44-s + 804.·46-s + 272.·47-s + ⋯ |
| L(s) = 1 | − 1.43·2-s + 1.05·4-s − 0.887·5-s − 0.0799·8-s + 1.27·10-s − 0.371·11-s + 0.395·13-s − 0.941·16-s − 1.33·17-s + 1.59·19-s − 0.936·20-s + 0.532·22-s − 1.79·23-s − 0.212·25-s − 0.567·26-s + 1.20·29-s − 0.486·31-s + 1.42·32-s + 1.91·34-s + 0.356·37-s − 2.28·38-s + 0.0709·40-s − 1.46·41-s − 1.40·43-s − 0.391·44-s + 2.57·46-s + 0.845·47-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
| \(L(2)\) |
\(\approx\) |
\(0.4901751783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4901751783\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 4.05T + 8T^{2} \) |
| 5 | \( 1 + 9.92T + 125T^{2} \) |
| 11 | \( 1 + 13.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 18.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 93.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 131.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 198.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 188.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 83.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 80.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 385.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 397.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 272.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 36.9T + 1.48e5T^{2} \) |
| 59 | \( 1 - 395.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 13.4T + 2.26e5T^{2} \) |
| 67 | \( 1 - 340.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 211.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 486.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 293.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 889.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.14e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.38e3T + 9.12e5T^{2} \) |
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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
−10.47188385876160828512850192738, −9.799264680449190772253419526571, −8.765698722765193006786492041902, −8.100042517380715567418808709306, −7.40621977890869333021944868174, −6.39718756061717913115349679072, −4.85817914300657666144091404028, −3.63580062224486940137302919919, −2.04149999653354838319977043441, −0.54263788537754972647402796842,
0.54263788537754972647402796842, 2.04149999653354838319977043441, 3.63580062224486940137302919919, 4.85817914300657666144091404028, 6.39718756061717913115349679072, 7.40621977890869333021944868174, 8.100042517380715567418808709306, 8.765698722765193006786492041902, 9.799264680449190772253419526571, 10.47188385876160828512850192738
