| L(s) = 1 | − 1.56·2-s + 0.438·4-s − 3.56·5-s − 0.561·7-s + 2.43·8-s + 5.56·10-s − 2·11-s + 0.876·14-s − 4.68·16-s + 1.56·17-s − 7.12·19-s − 1.56·20-s + 3.12·22-s − 2·23-s + 7.68·25-s − 0.246·28-s − 6.68·29-s − 2.56·31-s + 2.43·32-s − 2.43·34-s + 2·35-s − 7.56·37-s + 11.1·38-s − 8.68·40-s − 1.56·41-s + 4.56·43-s − 0.876·44-s + ⋯ |
| L(s) = 1 | − 1.10·2-s + 0.219·4-s − 1.59·5-s − 0.212·7-s + 0.862·8-s + 1.75·10-s − 0.603·11-s + 0.234·14-s − 1.17·16-s + 0.378·17-s − 1.63·19-s − 0.349·20-s + 0.665·22-s − 0.417·23-s + 1.53·25-s − 0.0465·28-s − 1.24·29-s − 0.460·31-s + 0.431·32-s − 0.418·34-s + 0.338·35-s − 1.24·37-s + 1.80·38-s − 1.37·40-s − 0.243·41-s + 0.695·43-s − 0.132·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2746666728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2746666728\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.56T + 2T^{2} \) |
| 5 | \( 1 + 3.56T + 5T^{2} \) |
| 7 | \( 1 + 0.561T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 - 1.56T + 17T^{2} \) |
| 19 | \( 1 + 7.12T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + 6.68T + 29T^{2} \) |
| 31 | \( 1 + 2.56T + 31T^{2} \) |
| 37 | \( 1 + 7.56T + 37T^{2} \) |
| 41 | \( 1 + 1.56T + 41T^{2} \) |
| 43 | \( 1 - 4.56T + 43T^{2} \) |
| 47 | \( 1 - 8.24T + 47T^{2} \) |
| 53 | \( 1 - 0.684T + 53T^{2} \) |
| 59 | \( 1 + 2.87T + 59T^{2} \) |
| 61 | \( 1 - 3.87T + 61T^{2} \) |
| 67 | \( 1 + 4.56T + 67T^{2} \) |
| 71 | \( 1 - 14T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 5.43T + 79T^{2} \) |
| 83 | \( 1 + 0.876T + 83T^{2} \) |
| 89 | \( 1 - 4.87T + 89T^{2} \) |
| 97 | \( 1 - 8.56T + 97T^{2} \) |
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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
−9.263489553812535001502362649945, −8.608679826875966097356259203068, −7.921174208219334450299661492516, −7.48121404536555195585443578199, −6.58540375912215168389480304264, −5.23574649215559206737936130433, −4.23872030791686903669152338881, −3.56797210881569497076903541704, −2.06902170765143710046164267312, −0.42231529706299085034984012268,
0.42231529706299085034984012268, 2.06902170765143710046164267312, 3.56797210881569497076903541704, 4.23872030791686903669152338881, 5.23574649215559206737936130433, 6.58540375912215168389480304264, 7.48121404536555195585443578199, 7.921174208219334450299661492516, 8.608679826875966097356259203068, 9.263489553812535001502362649945
