| L(s) = 1 | + (2.98 + 0.323i)3-s + 2.23i·5-s − 4.72·7-s + (8.79 + 1.92i)9-s − 4.76i·11-s + 1.06·13-s + (−0.722 + 6.66i)15-s + 26.7i·17-s − 8.12·19-s + (−14.0 − 1.52i)21-s + 40.0i·23-s − 5.00·25-s + (25.5 + 8.59i)27-s − 20.8i·29-s + 33.7·31-s + ⋯ |
| L(s) = 1 | + (0.994 + 0.107i)3-s + 0.447i·5-s − 0.675·7-s + (0.976 + 0.214i)9-s − 0.433i·11-s + 0.0820·13-s + (−0.0481 + 0.444i)15-s + 1.57i·17-s − 0.427·19-s + (−0.671 − 0.0727i)21-s + 1.74i·23-s − 0.200·25-s + (0.948 + 0.318i)27-s − 0.719i·29-s + 1.08·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.107 - 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.107 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.288019397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.288019397\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.98 - 0.323i)T \) |
| 5 | \( 1 - 2.23iT \) |
| good | 7 | \( 1 + 4.72T + 49T^{2} \) |
| 11 | \( 1 + 4.76iT - 121T^{2} \) |
| 13 | \( 1 - 1.06T + 169T^{2} \) |
| 17 | \( 1 - 26.7iT - 289T^{2} \) |
| 19 | \( 1 + 8.12T + 361T^{2} \) |
| 23 | \( 1 - 40.0iT - 529T^{2} \) |
| 29 | \( 1 + 20.8iT - 841T^{2} \) |
| 31 | \( 1 - 33.7T + 961T^{2} \) |
| 37 | \( 1 - 60.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 59.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 56.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 9.68iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 93.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 17.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 57.7T + 3.72e3T^{2} \) |
| 67 | \( 1 - 101.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 90.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 40.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 65.3T + 6.24e3T^{2} \) |
| 83 | \( 1 + 117. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 119. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 15.2T + 9.40e3T^{2} \) |
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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
−9.859467566129979707261305086749, −9.322156645269341027393024304184, −8.207421886260119502868848218114, −7.79090011652037464628563496806, −6.56856040866447867483781711491, −5.97105900954805690857038451977, −4.43737494257539905405475003229, −3.55323174505918763239140945258, −2.79068170335096752675711131244, −1.49507075434495738751739398461,
0.64717239491588020281933019280, 2.21699917570103370135690221037, 3.07697776011677997775381028876, 4.23937442853606557423947636893, 5.04097759200535353706030457695, 6.51959792986788947814157565653, 7.05472578165613230443662455877, 8.165851457498974040999085885728, 8.746162054070440407433074301018, 9.665712853016213279442661915603
