L(s) = 1 | − 4.42i·2-s − 11.6·4-s + 20.3i·5-s + 15.9i·8-s + 90.2·10-s + 70.0i·11-s + 46.8·13-s − 22.1·16-s − 236. i·20-s + 310.·22-s − 290.·25-s − 207. i·26-s + 225. i·32-s − 325.·40-s + 486. i·41-s + ⋯ |
L(s) = 1 | − 1.56i·2-s − 1.45·4-s + 1.82i·5-s + 0.705i·8-s + 2.85·10-s + 1.91i·11-s + 1.00·13-s − 0.346·16-s − 2.64i·20-s + 3.00·22-s − 2.32·25-s − 1.56i·26-s + 1.24i·32-s − 1.28·40-s + 1.85i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.33582\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.33582\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 - 46.8T \) |
good | 2 | \( 1 + 4.42iT - 8T^{2} \) |
| 5 | \( 1 - 20.3iT - 125T^{2} \) |
| 7 | \( 1 - 343T^{2} \) |
| 11 | \( 1 - 70.0iT - 1.33e3T^{2} \) |
| 17 | \( 1 + 4.91e3T^{2} \) |
| 19 | \( 1 - 6.85e3T^{2} \) |
| 23 | \( 1 + 1.21e4T^{2} \) |
| 29 | \( 1 + 2.43e4T^{2} \) |
| 31 | \( 1 - 2.97e4T^{2} \) |
| 37 | \( 1 - 5.06e4T^{2} \) |
| 41 | \( 1 - 486. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 452T + 7.95e4T^{2} \) |
| 47 | \( 1 + 71.1iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 1.48e5T^{2} \) |
| 59 | \( 1 + 696. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 944.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 3.00e5T^{2} \) |
| 71 | \( 1 - 123. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 3.89e5T^{2} \) |
| 79 | \( 1 - 418.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.50e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 155. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 9.12e5T^{2} \) |
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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
−12.80056317237265360561319787638, −11.78041628046654923804559932267, −10.91563164119839698012315134805, −10.24948645411412899446831703633, −9.408438562124640929267531943094, −7.52620405642343953344572689283, −6.43930566248715737132062517212, −4.28916520528875930588625783628, −3.07536055520435076079127533659, −1.93414730410789057295757879670,
0.74947780079364504799971857884, 4.08861906623796825448220164795, 5.46765207885993771344812385107, 6.02761560678818687588752145036, 7.74799160357728494602927486165, 8.735400857083690430728815360239, 8.973774865918991405321458958340, 11.00512185005866317058264490065, 12.33211395464503882824160919616, 13.59137040171676956124902723077