| L(s) = 1 | + 1.11·2-s + 3-s − 0.753·4-s − 0.900·5-s + 1.11·6-s − 3.07·8-s + 9-s − 1.00·10-s − 4.69·11-s − 0.753·12-s + 1.15·13-s − 0.900·15-s − 1.92·16-s + 5.04·17-s + 1.11·18-s + 0.849·19-s + 0.678·20-s − 5.24·22-s − 1.63·23-s − 3.07·24-s − 4.18·25-s + 1.29·26-s + 27-s + 7.97·29-s − 1.00·30-s + 2.94·31-s + 3.99·32-s + ⋯ |
| L(s) = 1 | + 0.789·2-s + 0.577·3-s − 0.376·4-s − 0.402·5-s + 0.455·6-s − 1.08·8-s + 0.333·9-s − 0.317·10-s − 1.41·11-s − 0.217·12-s + 0.320·13-s − 0.232·15-s − 0.481·16-s + 1.22·17-s + 0.263·18-s + 0.194·19-s + 0.151·20-s − 1.11·22-s − 0.341·23-s − 0.627·24-s − 0.837·25-s + 0.253·26-s + 0.192·27-s + 1.48·29-s − 0.183·30-s + 0.528·31-s + 0.706·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.346188099\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.346188099\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 1.11T + 2T^{2} \) |
| 5 | \( 1 + 0.900T + 5T^{2} \) |
| 11 | \( 1 + 4.69T + 11T^{2} \) |
| 13 | \( 1 - 1.15T + 13T^{2} \) |
| 17 | \( 1 - 5.04T + 17T^{2} \) |
| 19 | \( 1 - 0.849T + 19T^{2} \) |
| 23 | \( 1 + 1.63T + 23T^{2} \) |
| 29 | \( 1 - 7.97T + 29T^{2} \) |
| 31 | \( 1 - 2.94T + 31T^{2} \) |
| 37 | \( 1 - 3.09T + 37T^{2} \) |
| 43 | \( 1 + 3.01T + 43T^{2} \) |
| 47 | \( 1 + 8.14T + 47T^{2} \) |
| 53 | \( 1 + 8.01T + 53T^{2} \) |
| 59 | \( 1 + 2.60T + 59T^{2} \) |
| 61 | \( 1 + 7.31T + 61T^{2} \) |
| 67 | \( 1 - 0.796T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 - 1.76T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 + 0.878T + 97T^{2} \) |
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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
−8.035193500745842961194266710262, −7.64634665265946892332986128049, −6.46548306484905289058772419132, −5.84289357447198328405712846251, −4.94774989968536907989345135768, −4.58198180824730107729173960505, −3.40546864452203665829897065242, −3.22659395793948903886128078255, −2.15180580173586111358887683559, −0.68060337519942704907039452933,
0.68060337519942704907039452933, 2.15180580173586111358887683559, 3.22659395793948903886128078255, 3.40546864452203665829897065242, 4.58198180824730107729173960505, 4.94774989968536907989345135768, 5.84289357447198328405712846251, 6.46548306484905289058772419132, 7.64634665265946892332986128049, 8.035193500745842961194266710262
