| L(s) = 1 | − 1.41i·2-s − 2.00·4-s − 0.589i·5-s − 3.15·7-s + 2.82i·8-s − 0.833·10-s − 9.38i·11-s + 12.2·13-s + 4.45i·14-s + 4.00·16-s + 28.3i·17-s − 22.4·19-s + 1.17i·20-s − 13.2·22-s − 10.0i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s − 0.117i·5-s − 0.450·7-s + 0.353i·8-s − 0.0833·10-s − 0.853i·11-s + 0.939·13-s + 0.318i·14-s + 0.250·16-s + 1.66i·17-s − 1.17·19-s + 0.0589i·20-s − 0.603·22-s − 0.437i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.438846502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.438846502\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 0.589iT - 25T^{2} \) |
| 7 | \( 1 + 3.15T + 49T^{2} \) |
| 11 | \( 1 + 9.38iT - 121T^{2} \) |
| 13 | \( 1 - 12.2T + 169T^{2} \) |
| 17 | \( 1 - 28.3iT - 289T^{2} \) |
| 19 | \( 1 + 22.4T + 361T^{2} \) |
| 23 | \( 1 + 10.0iT - 529T^{2} \) |
| 29 | \( 1 - 24.0iT - 841T^{2} \) |
| 31 | \( 1 + 45.4T + 961T^{2} \) |
| 37 | \( 1 - 31.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 70.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 14.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 55.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 25.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 28.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 113.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 50.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 8.77iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 23.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 132.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 67.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 72.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 157.T + 9.40e3T^{2} \) |
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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
−9.302705205065056171966315989244, −8.526800945680272487566451660429, −8.168596304264759261091712993550, −6.63441501924639036909862771740, −6.12044057673371334387601414826, −5.06956105860214646569182316522, −3.90116021395459799261367875647, −3.37516057797265839630333079647, −2.08756476468498690097528654399, −0.940938014598044732413071589097,
0.50619688218407664804053777712, 2.14380287870122546436372983843, 3.40011536886970913730143861660, 4.36577531628789120151368123607, 5.24305744217178257583950306486, 6.17841647564885302023106813805, 6.93480516309525032069332354432, 7.52181051194335801009442542334, 8.550255336818476996399552360242, 9.255124765875452977460008835492
