| L(s) = 1 | + 2-s + 3.14·3-s + 4-s − 5-s + 3.14·6-s − 3.89·7-s + 8-s + 6.89·9-s − 10-s − 4.29·11-s + 3.14·12-s + 4.34·13-s − 3.89·14-s − 3.14·15-s + 16-s + 1.60·17-s + 6.89·18-s − 1.20·19-s − 20-s − 12.2·21-s − 4.29·22-s − 8.34·23-s + 3.14·24-s + 25-s + 4.34·26-s + 12.2·27-s − 3.89·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.81·3-s + 0.5·4-s − 0.447·5-s + 1.28·6-s − 1.47·7-s + 0.353·8-s + 2.29·9-s − 0.316·10-s − 1.29·11-s + 0.907·12-s + 1.20·13-s − 1.04·14-s − 0.812·15-s + 0.250·16-s + 0.388·17-s + 1.62·18-s − 0.275·19-s − 0.223·20-s − 2.67·21-s − 0.914·22-s − 1.74·23-s + 0.641·24-s + 0.200·25-s + 0.852·26-s + 2.35·27-s − 0.735·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.693506983\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.693506983\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 3.14T + 3T^{2} \) |
| 7 | \( 1 + 3.89T + 7T^{2} \) |
| 11 | \( 1 + 4.29T + 11T^{2} \) |
| 13 | \( 1 - 4.34T + 13T^{2} \) |
| 17 | \( 1 - 1.60T + 17T^{2} \) |
| 19 | \( 1 + 1.20T + 19T^{2} \) |
| 23 | \( 1 + 8.34T + 23T^{2} \) |
| 31 | \( 1 + 2.39T + 31T^{2} \) |
| 37 | \( 1 + 9.78T + 37T^{2} \) |
| 41 | \( 1 - 5.78T + 41T^{2} \) |
| 43 | \( 1 - 3.60T + 43T^{2} \) |
| 47 | \( 1 - 8.58T + 47T^{2} \) |
| 53 | \( 1 - 4.80T + 53T^{2} \) |
| 59 | \( 1 + 4.05T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 - 9.08T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 2.39T + 73T^{2} \) |
| 79 | \( 1 - 7.55T + 79T^{2} \) |
| 83 | \( 1 - 2.79T + 83T^{2} \) |
| 89 | \( 1 + 4.29T + 89T^{2} \) |
| 97 | \( 1 + 0.348T + 97T^{2} \) |
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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
−12.30944438702545142400179855434, −10.63558349084430931817762020080, −9.903128644917207584138040105758, −8.807995506409440880056456620375, −7.965254244668962494130985274267, −7.08820154662815520981181126867, −5.83393683358686715852853527128, −3.99536896728196627920557218645, −3.40347079805317736636773815143, −2.35928655279984995783914105728,
2.35928655279984995783914105728, 3.40347079805317736636773815143, 3.99536896728196627920557218645, 5.83393683358686715852853527128, 7.08820154662815520981181126867, 7.965254244668962494130985274267, 8.807995506409440880056456620375, 9.903128644917207584138040105758, 10.63558349084430931817762020080, 12.30944438702545142400179855434
