| L(s) = 1 | + 4.69·3-s + 7.54i·7-s + 13.0·9-s + (5.78 − 9.35i)11-s + 0.829i·13-s + 29.8i·17-s + 36.2i·19-s + 35.4i·21-s − 29.7·23-s + 18.9·27-s − 10.0i·29-s + 34.6·31-s + (27.1 − 43.9i)33-s + 61.9·37-s + 3.89i·39-s + ⋯ |
| L(s) = 1 | + 1.56·3-s + 1.07i·7-s + 1.44·9-s + (0.525 − 0.850i)11-s + 0.0637i·13-s + 1.75i·17-s + 1.90i·19-s + 1.68i·21-s − 1.29·23-s + 0.700·27-s − 0.346i·29-s + 1.11·31-s + (0.822 − 1.33i)33-s + 1.67·37-s + 0.0997i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.405030836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.405030836\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (-5.78 + 9.35i)T \) |
| good | 3 | \( 1 - 4.69T + 9T^{2} \) |
| 7 | \( 1 - 7.54iT - 49T^{2} \) |
| 13 | \( 1 - 0.829iT - 169T^{2} \) |
| 17 | \( 1 - 29.8iT - 289T^{2} \) |
| 19 | \( 1 - 36.2iT - 361T^{2} \) |
| 23 | \( 1 + 29.7T + 529T^{2} \) |
| 29 | \( 1 + 10.0iT - 841T^{2} \) |
| 31 | \( 1 - 34.6T + 961T^{2} \) |
| 37 | \( 1 - 61.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + 11.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 39.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 6.35T + 2.20e3T^{2} \) |
| 53 | \( 1 + 56.5T + 2.80e3T^{2} \) |
| 59 | \( 1 - 70.4T + 3.48e3T^{2} \) |
| 61 | \( 1 + 8.41iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 18.7T + 4.48e3T^{2} \) |
| 71 | \( 1 + 3.79T + 5.04e3T^{2} \) |
| 73 | \( 1 - 70.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 127. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 100. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 66.0T + 7.92e3T^{2} \) |
| 97 | \( 1 - 1.65T + 9.40e3T^{2} \) |
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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
−9.705652759665628991915059269567, −8.682977282249078533596787014872, −8.325412761761573987477720150535, −7.78897565066826125104029262124, −6.22944399944748225866234557209, −5.83006024690454170948207693315, −4.10010743511916743605965548788, −3.56789978794398547822969226693, −2.43639703970402331620115489386, −1.60680851197605165580451532809,
0.867216536749181045205221735827, 2.31598965217613308578979128802, 3.07383584916510838648411653321, 4.23773557945822451540322425458, 4.75015197089686000094157482609, 6.54036936329862126912362056742, 7.27887156871006440830550861442, 7.79070791011121019462754302621, 8.763511680237432432758073423544, 9.664781228716987670152883511595
