| L(s) = 1 | − 2.65·2-s + 7.25·3-s − 0.955·4-s + 11.9·5-s − 19.2·6-s + 3.65·7-s + 23.7·8-s + 25.6·9-s − 31.6·10-s − 6.93·12-s − 13·13-s − 9.70·14-s + 86.5·15-s − 55.4·16-s − 118.·17-s − 68.0·18-s − 69.1·19-s − 11.3·20-s + 26.5·21-s + 208.·23-s + 172.·24-s + 17.1·25-s + 34.5·26-s − 9.77·27-s − 3.49·28-s − 39.8·29-s − 229.·30-s + ⋯ |
| L(s) = 1 | − 0.938·2-s + 1.39·3-s − 0.119·4-s + 1.06·5-s − 1.31·6-s + 0.197·7-s + 1.05·8-s + 0.950·9-s − 1.00·10-s − 0.166·12-s − 0.277·13-s − 0.185·14-s + 1.48·15-s − 0.866·16-s − 1.68·17-s − 0.891·18-s − 0.835·19-s − 0.127·20-s + 0.275·21-s + 1.89·23-s + 1.46·24-s + 0.136·25-s + 0.260·26-s − 0.0696·27-s − 0.0235·28-s − 0.255·29-s − 1.39·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 2 | \( 1 + 2.65T + 8T^{2} \) |
| 3 | \( 1 - 7.25T + 27T^{2} \) |
| 5 | \( 1 - 11.9T + 125T^{2} \) |
| 7 | \( 1 - 3.65T + 343T^{2} \) |
| 17 | \( 1 + 118.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 69.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 208.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 39.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 104.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 337.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 25.9T + 6.89e4T^{2} \) |
| 43 | \( 1 + 366.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 274.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 84.7T + 1.48e5T^{2} \) |
| 59 | \( 1 - 50.5T + 2.05e5T^{2} \) |
| 61 | \( 1 + 213.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 420.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 276.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.16e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 23.3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 731.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 494.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.55e3T + 9.12e5T^{2} \) |
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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
−8.791428427119198685383207193211, −8.298490005173625793777526361999, −7.24693248708301401499334185436, −6.62533442116725893171964089529, −5.22370766126580175653781522182, −4.42798107966459979268237825489, −3.23631823035476314652170209656, −2.12027142902232956169652244065, −1.63992131642716016208614571673, 0,
1.63992131642716016208614571673, 2.12027142902232956169652244065, 3.23631823035476314652170209656, 4.42798107966459979268237825489, 5.22370766126580175653781522182, 6.62533442116725893171964089529, 7.24693248708301401499334185436, 8.298490005173625793777526361999, 8.791428427119198685383207193211
