| L(s) = 1 | + 2-s − 1.10·3-s + 4-s + 3.90·5-s − 1.10·6-s − 4.77·7-s + 8-s − 1.76·9-s + 3.90·10-s + 5.69·11-s − 1.10·12-s − 4.09·13-s − 4.77·14-s − 4.32·15-s + 16-s − 0.265·17-s − 1.76·18-s − 19-s + 3.90·20-s + 5.30·21-s + 5.69·22-s − 3.37·23-s − 1.10·24-s + 10.2·25-s − 4.09·26-s + 5.29·27-s − 4.77·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.640·3-s + 0.5·4-s + 1.74·5-s − 0.453·6-s − 1.80·7-s + 0.353·8-s − 0.589·9-s + 1.23·10-s + 1.71·11-s − 0.320·12-s − 1.13·13-s − 1.27·14-s − 1.11·15-s + 0.250·16-s − 0.0642·17-s − 0.416·18-s − 0.229·19-s + 0.872·20-s + 1.15·21-s + 1.21·22-s − 0.703·23-s − 0.226·24-s + 2.04·25-s − 0.802·26-s + 1.01·27-s − 0.902·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.690022082\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.690022082\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 1.10T + 3T^{2} \) |
| 5 | \( 1 - 3.90T + 5T^{2} \) |
| 7 | \( 1 + 4.77T + 7T^{2} \) |
| 11 | \( 1 - 5.69T + 11T^{2} \) |
| 13 | \( 1 + 4.09T + 13T^{2} \) |
| 17 | \( 1 + 0.265T + 17T^{2} \) |
| 23 | \( 1 + 3.37T + 23T^{2} \) |
| 29 | \( 1 + 8.30T + 29T^{2} \) |
| 31 | \( 1 + 6.89T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 + 2.35T + 43T^{2} \) |
| 47 | \( 1 - 0.906T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 - 14.5T + 59T^{2} \) |
| 61 | \( 1 - 3.09T + 61T^{2} \) |
| 67 | \( 1 - 14.6T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 + 4.58T + 73T^{2} \) |
| 79 | \( 1 + 5.66T + 79T^{2} \) |
| 83 | \( 1 - 9.13T + 83T^{2} \) |
| 89 | \( 1 + 6.19T + 89T^{2} \) |
| 97 | \( 1 - 8.89T + 97T^{2} \) |
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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
−7.36606572991955156009990062849, −6.76659905829728764914327725996, −6.25342536768827627440432988850, −5.78410274165270585251017738397, −5.44265068628264663332667527781, −4.25864328314813247567229568215, −3.56690435135525439809899062188, −2.58638612647275465261959401816, −2.06804289486049022147777561890, −0.72665901864605275519388150901,
0.72665901864605275519388150901, 2.06804289486049022147777561890, 2.58638612647275465261959401816, 3.56690435135525439809899062188, 4.25864328314813247567229568215, 5.44265068628264663332667527781, 5.78410274165270585251017738397, 6.25342536768827627440432988850, 6.76659905829728764914327725996, 7.36606572991955156009990062849
