L(s) = 1 | − 2-s − 0.732·3-s + 4-s + 0.732·6-s + 3·7-s − 8-s − 2.46·9-s + 3·11-s − 0.732·12-s − 3·14-s + 16-s + 8.19·17-s + 2.46·18-s − 0.464·19-s − 2.19·21-s − 3·22-s + 9.46·23-s + 0.732·24-s + 4·27-s + 3·28-s − 2.53·29-s + 4.73·31-s − 32-s − 2.19·33-s − 8.19·34-s − 2.46·36-s + 0.803·37-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.422·3-s + 0.5·4-s + 0.298·6-s + 1.13·7-s − 0.353·8-s − 0.821·9-s + 0.904·11-s − 0.211·12-s − 0.801·14-s + 0.250·16-s + 1.98·17-s + 0.580·18-s − 0.106·19-s − 0.479·21-s − 0.639·22-s + 1.97·23-s + 0.149·24-s + 0.769·27-s + 0.566·28-s − 0.470·29-s + 0.849·31-s − 0.176·32-s − 0.382·33-s − 1.40·34-s − 0.410·36-s + 0.132·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.788888684\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.788888684\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + 0.732T + 3T^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 17 | \( 1 - 8.19T + 17T^{2} \) |
| 19 | \( 1 + 0.464T + 19T^{2} \) |
| 23 | \( 1 - 9.46T + 23T^{2} \) |
| 29 | \( 1 + 2.53T + 29T^{2} \) |
| 31 | \( 1 - 4.73T + 31T^{2} \) |
| 37 | \( 1 - 0.803T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 3T + 47T^{2} \) |
| 53 | \( 1 + 0.464T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 6.19T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 + 4.19T + 79T^{2} \) |
| 83 | \( 1 - 8.19T + 83T^{2} \) |
| 89 | \( 1 - 6.80T + 89T^{2} \) |
| 97 | \( 1 + 9.12T + 97T^{2} \) |
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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
−7.935001634114670014301749078611, −7.22166928412123781669066765499, −6.47675308056408913045695037568, −5.67278254943997710947392829873, −5.22813989070521753434387550577, −4.34656374736070567874193758043, −3.31913045626660284087648110089, −2.59046403397414039205170155745, −1.34962702958601886699772344905, −0.873919183151157023377589516158,
0.873919183151157023377589516158, 1.34962702958601886699772344905, 2.59046403397414039205170155745, 3.31913045626660284087648110089, 4.34656374736070567874193758043, 5.22813989070521753434387550577, 5.67278254943997710947392829873, 6.47675308056408913045695037568, 7.22166928412123781669066765499, 7.935001634114670014301749078611