| L(s) = 1 | + 1.73i·3-s − 6.27·5-s − 12.2·7-s − 2.99·9-s − 0.274·11-s − 13.0i·13-s − 10.8i·15-s + 17.3·17-s + (−7.54 − 17.4i)19-s − 21.2i·21-s − 20.5·23-s + 14.3·25-s − 5.19i·27-s + 26.0i·29-s + 23.5i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 1.25·5-s − 1.75·7-s − 0.333·9-s − 0.0249·11-s − 1.00i·13-s − 0.724i·15-s + 1.02·17-s + (−0.397 − 0.917i)19-s − 1.01i·21-s − 0.893·23-s + 0.574·25-s − 0.192i·27-s + 0.897i·29-s + 0.758i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7242911910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7242911910\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 19 | \( 1 + (7.54 + 17.4i)T \) |
| good | 5 | \( 1 + 6.27T + 25T^{2} \) |
| 7 | \( 1 + 12.2T + 49T^{2} \) |
| 11 | \( 1 + 0.274T + 121T^{2} \) |
| 13 | \( 1 + 13.0iT - 169T^{2} \) |
| 17 | \( 1 - 17.3T + 289T^{2} \) |
| 23 | \( 1 + 20.5T + 529T^{2} \) |
| 29 | \( 1 - 26.0iT - 841T^{2} \) |
| 31 | \( 1 - 23.5iT - 961T^{2} \) |
| 37 | \( 1 - 66.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 3.57iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 48.1T + 1.84e3T^{2} \) |
| 47 | \( 1 - 12.4T + 2.20e3T^{2} \) |
| 53 | \( 1 + 25.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 0.230iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 28.8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 102. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 107. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 11.1T + 5.32e3T^{2} \) |
| 79 | \( 1 - 26.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 89.6T + 6.88e3T^{2} \) |
| 89 | \( 1 + 139. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 41.3iT - 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.07697435670020944779865129821, −9.169420261059373023474591034717, −8.303602849960947725547640121448, −7.44785881255003039094879013000, −6.54550857791459342672566107415, −5.58965919621856108641562182641, −4.44064625875485112400440518318, −3.40442779033803343925674683678, −3.01688544031175748496630072022, −0.51719715638428225606137700611,
0.51083141874884933090068115775, 2.32694747372567297771738071452, 3.63266478891788088794425443622, 4.05826419893936350279781215799, 5.81791160511922618108324360551, 6.40544754181908157954014954085, 7.43737747081130714507149610516, 7.88773667145422311497716371138, 9.054636291235414194944924960849, 9.741241355856786147003631136199
