L(s) = 1 | + 0.745·2-s − 2.05·3-s − 1.44·4-s − 5-s − 1.52·6-s − 3.86·7-s − 2.56·8-s + 1.20·9-s − 0.745·10-s + 11-s + 2.96·12-s − 2.04·13-s − 2.88·14-s + 2.05·15-s + 0.975·16-s − 3.70·17-s + 0.898·18-s − 19-s + 1.44·20-s + 7.93·21-s + 0.745·22-s + 2.09·23-s + 5.26·24-s + 25-s − 1.52·26-s + 3.68·27-s + 5.58·28-s + ⋯ |
L(s) = 1 | + 0.527·2-s − 1.18·3-s − 0.722·4-s − 0.447·5-s − 0.623·6-s − 1.46·7-s − 0.907·8-s + 0.401·9-s − 0.235·10-s + 0.301·11-s + 0.855·12-s − 0.566·13-s − 0.770·14-s + 0.529·15-s + 0.243·16-s − 0.898·17-s + 0.211·18-s − 0.229·19-s + 0.323·20-s + 1.73·21-s + 0.158·22-s + 0.436·23-s + 1.07·24-s + 0.200·25-s − 0.298·26-s + 0.708·27-s + 1.05·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4019969675\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4019969675\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 2 | \( 1 - 0.745T + 2T^{2} \) |
| 3 | \( 1 + 2.05T + 3T^{2} \) |
| 7 | \( 1 + 3.86T + 7T^{2} \) |
| 13 | \( 1 + 2.04T + 13T^{2} \) |
| 17 | \( 1 + 3.70T + 17T^{2} \) |
| 23 | \( 1 - 2.09T + 23T^{2} \) |
| 29 | \( 1 + 1.56T + 29T^{2} \) |
| 31 | \( 1 - 5.38T + 31T^{2} \) |
| 37 | \( 1 + 7.65T + 37T^{2} \) |
| 41 | \( 1 - 2.55T + 41T^{2} \) |
| 43 | \( 1 - 1.43T + 43T^{2} \) |
| 47 | \( 1 + 5.29T + 47T^{2} \) |
| 53 | \( 1 - 2.10T + 53T^{2} \) |
| 59 | \( 1 - 0.643T + 59T^{2} \) |
| 61 | \( 1 - 14.3T + 61T^{2} \) |
| 67 | \( 1 + 7.38T + 67T^{2} \) |
| 71 | \( 1 + 3.12T + 71T^{2} \) |
| 73 | \( 1 - 6.41T + 73T^{2} \) |
| 79 | \( 1 + 3.06T + 79T^{2} \) |
| 83 | \( 1 - 6.28T + 83T^{2} \) |
| 89 | \( 1 + 1.85T + 89T^{2} \) |
| 97 | \( 1 - 6.58T + 97T^{2} \) |
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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
−9.955703209724858679266285718266, −9.217420595500690627263202287707, −8.399328770267622514441581515719, −6.95188881553273920899195999179, −6.44278004861346083131231774367, −5.57588487317260467838795274060, −4.74948196584749741372996000361, −3.87670641826391346368223502769, −2.86605906921975882889489991889, −0.45573796087617507196452614855,
0.45573796087617507196452614855, 2.86605906921975882889489991889, 3.87670641826391346368223502769, 4.74948196584749741372996000361, 5.57588487317260467838795274060, 6.44278004861346083131231774367, 6.95188881553273920899195999179, 8.399328770267622514441581515719, 9.217420595500690627263202287707, 9.955703209724858679266285718266