| L(s) = 1 | + 3-s + 2.75·5-s − 7-s + 9-s − 4.32·11-s + 1.57·13-s + 2.75·15-s + 0.819·17-s + 2.81·19-s − 21-s − 23-s + 2.57·25-s + 27-s − 0.819·29-s − 8.32·31-s − 4.32·33-s − 2.75·35-s + 8.68·37-s + 1.57·39-s + 10.3·41-s + 0.429·43-s + 2.75·45-s + 10.1·47-s + 49-s + 0.819·51-s − 7.39·53-s − 11.8·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.23·5-s − 0.377·7-s + 0.333·9-s − 1.30·11-s + 0.435·13-s + 0.710·15-s + 0.198·17-s + 0.646·19-s − 0.218·21-s − 0.208·23-s + 0.514·25-s + 0.192·27-s − 0.152·29-s − 1.49·31-s − 0.752·33-s − 0.465·35-s + 1.42·37-s + 0.251·39-s + 1.61·41-s + 0.0654·43-s + 0.410·45-s + 1.48·47-s + 0.142·49-s + 0.114·51-s − 1.01·53-s − 1.60·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.070810146\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.070810146\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 2.75T + 5T^{2} \) |
| 11 | \( 1 + 4.32T + 11T^{2} \) |
| 13 | \( 1 - 1.57T + 13T^{2} \) |
| 17 | \( 1 - 0.819T + 17T^{2} \) |
| 19 | \( 1 - 2.81T + 19T^{2} \) |
| 29 | \( 1 + 0.819T + 29T^{2} \) |
| 31 | \( 1 + 8.32T + 31T^{2} \) |
| 37 | \( 1 - 8.68T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 - 0.429T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 + 7.39T + 53T^{2} \) |
| 59 | \( 1 - 6.39T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 - 1.60T + 67T^{2} \) |
| 71 | \( 1 - 2.06T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 - 6.81T + 79T^{2} \) |
| 83 | \( 1 - 4.32T + 83T^{2} \) |
| 89 | \( 1 - 7.93T + 89T^{2} \) |
| 97 | \( 1 - 1.67T + 97T^{2} \) |
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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
−7.73232041428886798594865576631, −7.38215624669794160660434191612, −6.35364537974451138304594311103, −5.71632095607667363932008038772, −5.29724511215228100678591657205, −4.24421396934927834986293557814, −3.37160319587115260599304289965, −2.55050074414854664675864156041, −2.03652969213474175099983400934, −0.846769875939663071435630144965,
0.846769875939663071435630144965, 2.03652969213474175099983400934, 2.55050074414854664675864156041, 3.37160319587115260599304289965, 4.24421396934927834986293557814, 5.29724511215228100678591657205, 5.71632095607667363932008038772, 6.35364537974451138304594311103, 7.38215624669794160660434191612, 7.73232041428886798594865576631
