L(s) = 1 | − 4.26·2-s + 10.1·4-s − 5·5-s + 30.7·7-s − 9.21·8-s + 21.3·10-s + 40.7·11-s + 63.2·13-s − 131.·14-s − 42.0·16-s − 6.58·17-s + 75.3·19-s − 50.8·20-s − 173.·22-s − 62.3·23-s + 25·25-s − 269.·26-s + 312.·28-s + 49.6·29-s + 103.·31-s + 252.·32-s + 28.0·34-s − 153.·35-s − 282.·37-s − 321.·38-s + 46.0·40-s − 157.·41-s + ⋯ |
L(s) = 1 | − 1.50·2-s + 1.27·4-s − 0.447·5-s + 1.66·7-s − 0.407·8-s + 0.673·10-s + 1.11·11-s + 1.34·13-s − 2.50·14-s − 0.656·16-s − 0.0940·17-s + 0.910·19-s − 0.568·20-s − 1.68·22-s − 0.565·23-s + 0.200·25-s − 2.03·26-s + 2.11·28-s + 0.317·29-s + 0.596·31-s + 1.39·32-s + 0.141·34-s − 0.742·35-s − 1.25·37-s − 1.37·38-s + 0.182·40-s − 0.600·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.167918062\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.167918062\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
good | 2 | \( 1 + 4.26T + 8T^{2} \) |
| 7 | \( 1 - 30.7T + 343T^{2} \) |
| 11 | \( 1 - 40.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 63.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 6.58T + 4.91e3T^{2} \) |
| 19 | \( 1 - 75.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 62.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 49.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 103.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 282.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 157.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 337.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 44.5T + 1.03e5T^{2} \) |
| 53 | \( 1 - 26.2T + 1.48e5T^{2} \) |
| 59 | \( 1 + 425.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 850.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 96.3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 952.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 50.8T + 3.89e5T^{2} \) |
| 79 | \( 1 + 197.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 197.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.36e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.43e3T + 9.12e5T^{2} \) |
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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
−10.81939904212213424802028127588, −9.842522722574301190127635484123, −8.658262277153445364084444410767, −8.407853250042967559280748385751, −7.47396556948554538029086414220, −6.48222691434272075392756449813, −4.97409491770830216879436917775, −3.76939229706880367161895096057, −1.77040465006059863744687634367, −0.984246551985948356398853228608,
0.984246551985948356398853228608, 1.77040465006059863744687634367, 3.76939229706880367161895096057, 4.97409491770830216879436917775, 6.48222691434272075392756449813, 7.47396556948554538029086414220, 8.407853250042967559280748385751, 8.658262277153445364084444410767, 9.842522722574301190127635484123, 10.81939904212213424802028127588