| L(s) = 1 | − 1.90i·2-s + 5.95i·3-s + 0.358·4-s + 11.3·6-s − 8.31i·8-s − 26.4·9-s − 17.4·11-s + 2.13i·12-s + 2.59i·13-s − 14.4·16-s + 50.4i·18-s − 19·19-s + 33.2i·22-s + 49.5·24-s + 4.94·26-s − 103. i·27-s + ⋯ |
| L(s) = 1 | − 0.954i·2-s + 1.98i·3-s + 0.0897·4-s + 1.89·6-s − 1.03i·8-s − 2.93·9-s − 1.58·11-s + 0.178i·12-s + 0.199i·13-s − 0.902·16-s + 2.80i·18-s − 19-s + 1.51i·22-s + 2.06·24-s + 0.190·26-s − 3.84i·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3000281290\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3000281290\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 + 1.90iT - 4T^{2} \) |
| 3 | \( 1 - 5.95iT - 9T^{2} \) |
| 7 | \( 1 + 49T^{2} \) |
| 11 | \( 1 + 17.4T + 121T^{2} \) |
| 13 | \( 1 - 2.59iT - 169T^{2} \) |
| 17 | \( 1 + 289T^{2} \) |
| 23 | \( 1 + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 58.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 + 1.84e3T^{2} \) |
| 47 | \( 1 + 2.20e3T^{2} \) |
| 53 | \( 1 + 7.48iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 + 17.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 48.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 + 89.5iT - 9.40e3T^{2} \) |
| 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
−11.02930692330647470576680923130, −10.29991970649422001741086847525, −9.971935136196316065467574492042, −8.931856589467002445583593065410, −7.997981133583423548505938635450, −6.32833148856528735033337994995, −5.16784536107453985191534549023, −4.33618515415743311627825533821, −3.28015214643935647321133160781, −2.46614804283359368974060994071,
0.10721341696309274739763426169, 1.96117785854221349495634961071, 2.79531318005581841953099743963, 5.21867310896249404570657433789, 5.96174034297558250596898369529, 6.74859626588733864392003048283, 7.60330638681485998775089835156, 8.015809830461047475326206050983, 8.839041413007453798629151540849, 10.66534407356422086162559280166
