| L(s) = 1 | + 1.67·2-s − 1.49·3-s + 0.798·4-s − 1.12·5-s − 2.49·6-s − 0.438·7-s − 2.00·8-s − 0.774·9-s − 1.88·10-s + 11-s − 1.19·12-s + 6.70·13-s − 0.733·14-s + 1.68·15-s − 4.95·16-s + 0.414·17-s − 1.29·18-s + 5.91·19-s − 0.900·20-s + 0.654·21-s + 1.67·22-s − 8.48·23-s + 2.99·24-s − 3.73·25-s + 11.2·26-s + 5.63·27-s − 0.350·28-s + ⋯ |
| L(s) = 1 | + 1.18·2-s − 0.861·3-s + 0.399·4-s − 0.503·5-s − 1.01·6-s − 0.165·7-s − 0.710·8-s − 0.258·9-s − 0.596·10-s + 0.301·11-s − 0.344·12-s + 1.86·13-s − 0.196·14-s + 0.433·15-s − 1.23·16-s + 0.100·17-s − 0.305·18-s + 1.35·19-s − 0.201·20-s + 0.142·21-s + 0.356·22-s − 1.76·23-s + 0.611·24-s − 0.746·25-s + 2.20·26-s + 1.08·27-s − 0.0661·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.827327675\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.827327675\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 2 | \( 1 - 1.67T + 2T^{2} \) |
| 3 | \( 1 + 1.49T + 3T^{2} \) |
| 5 | \( 1 + 1.12T + 5T^{2} \) |
| 7 | \( 1 + 0.438T + 7T^{2} \) |
| 13 | \( 1 - 6.70T + 13T^{2} \) |
| 17 | \( 1 - 0.414T + 17T^{2} \) |
| 19 | \( 1 - 5.91T + 19T^{2} \) |
| 23 | \( 1 + 8.48T + 23T^{2} \) |
| 29 | \( 1 - 6.46T + 29T^{2} \) |
| 31 | \( 1 - 8.15T + 31T^{2} \) |
| 37 | \( 1 + 0.0703T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 - 4.47T + 43T^{2} \) |
| 47 | \( 1 - 4.80T + 47T^{2} \) |
| 53 | \( 1 - 14.4T + 53T^{2} \) |
| 59 | \( 1 - 3.77T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 - 4.98T + 67T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 + 2.49T + 79T^{2} \) |
| 83 | \( 1 - 9.29T + 83T^{2} \) |
| 89 | \( 1 + 0.995T + 89T^{2} \) |
| 97 | \( 1 + 0.687T + 97T^{2} \) |
| 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
−9.603495058664453306930652848733, −8.621785577153940508301993624857, −7.903030597667126670333792983299, −6.57485100677451811236379600085, −6.00336765232754835535396104220, −5.52445189187851561973032716575, −4.34917168316029258390957037944, −3.80040391102369301946148272185, −2.80252673392701281629844977525, −0.863194794243108129276861927492,
0.863194794243108129276861927492, 2.80252673392701281629844977525, 3.80040391102369301946148272185, 4.34917168316029258390957037944, 5.52445189187851561973032716575, 6.00336765232754835535396104220, 6.57485100677451811236379600085, 7.903030597667126670333792983299, 8.621785577153940508301993624857, 9.603495058664453306930652848733
