| L(s) = 1 | + 6.04·5-s + 0.880·7-s + 12.2·11-s − 7.16i·13-s + 15.7·17-s + (18.2 + 5.29i)19-s + 19.1·23-s + 11.5·25-s − 51.7i·29-s + 32.5i·31-s + 5.32·35-s − 25.3i·37-s + 19.0i·41-s − 39.5·43-s + 24.4·47-s + ⋯ |
| L(s) = 1 | + 1.20·5-s + 0.125·7-s + 1.11·11-s − 0.551i·13-s + 0.926·17-s + (0.960 + 0.278i)19-s + 0.832·23-s + 0.460·25-s − 1.78i·29-s + 1.04i·31-s + 0.152·35-s − 0.686i·37-s + 0.465i·41-s − 0.920·43-s + 0.520·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.278i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.960 + 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.373889270\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.373889270\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-18.2 - 5.29i)T \) |
| good | 5 | \( 1 - 6.04T + 25T^{2} \) |
| 7 | \( 1 - 0.880T + 49T^{2} \) |
| 11 | \( 1 - 12.2T + 121T^{2} \) |
| 13 | \( 1 + 7.16iT - 169T^{2} \) |
| 17 | \( 1 - 15.7T + 289T^{2} \) |
| 23 | \( 1 - 19.1T + 529T^{2} \) |
| 29 | \( 1 + 51.7iT - 841T^{2} \) |
| 31 | \( 1 - 32.5iT - 961T^{2} \) |
| 37 | \( 1 + 25.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 19.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 39.5T + 1.84e3T^{2} \) |
| 47 | \( 1 - 24.4T + 2.20e3T^{2} \) |
| 53 | \( 1 + 13.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 65.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 22.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 18.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 76.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 85.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 150. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 29.0T + 6.88e3T^{2} \) |
| 89 | \( 1 + 127. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 127. iT - 9.40e3T^{2} \) |
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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
−8.717569082776862891426999097772, −7.86018431325808410315481013766, −7.03681221461684044124521917335, −6.18232082711343210798930349625, −5.62944963060262314920082720867, −4.86370333063031706357248415102, −3.71576830661232917164412669944, −2.86650168344556855983715389488, −1.73083885872653348647092240246, −0.931980881400707529899111194723,
1.09686150668029726456134165598, 1.76905164551641534438823148900, 2.95461361570833808379759052339, 3.81790728331825857301720045721, 5.01216752424356062745487550596, 5.50311972712702261042132710572, 6.50653305551148742670148767355, 6.92780368294275993024453162556, 7.950025062016677351578043945361, 8.908285774276250784841299704481
