| L(s) = 1 | + 2.26·2-s + 0.319·3-s − 2.86·4-s + 1.59·5-s + 0.723·6-s + 11.1·7-s − 24.6·8-s − 26.8·9-s + 3.62·10-s + 15.3·11-s − 0.913·12-s + 49.6·13-s + 25.2·14-s + 0.510·15-s − 32.9·16-s − 4.98·17-s − 60.9·18-s − 4.67·19-s − 4.57·20-s + 3.55·21-s + 34.7·22-s + 23·23-s − 7.85·24-s − 122.·25-s + 112.·26-s − 17.2·27-s − 31.9·28-s + ⋯ |
| L(s) = 1 | + 0.801·2-s + 0.0614·3-s − 0.357·4-s + 0.142·5-s + 0.0492·6-s + 0.601·7-s − 1.08·8-s − 0.996·9-s + 0.114·10-s + 0.420·11-s − 0.0219·12-s + 1.05·13-s + 0.482·14-s + 0.00878·15-s − 0.514·16-s − 0.0711·17-s − 0.798·18-s − 0.0564·19-s − 0.0511·20-s + 0.0369·21-s + 0.336·22-s + 0.208·23-s − 0.0668·24-s − 0.979·25-s + 0.849·26-s − 0.122·27-s − 0.215·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 23 | \( 1 - 23T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 - 2.26T + 8T^{2} \) |
| 3 | \( 1 - 0.319T + 27T^{2} \) |
| 5 | \( 1 - 1.59T + 125T^{2} \) |
| 7 | \( 1 - 11.1T + 343T^{2} \) |
| 11 | \( 1 - 15.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 49.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 4.98T + 4.91e3T^{2} \) |
| 19 | \( 1 + 4.67T + 6.85e3T^{2} \) |
| 31 | \( 1 + 300.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 353.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 150.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 371.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 179.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 80.4T + 1.48e5T^{2} \) |
| 59 | \( 1 + 402.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 110.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 611.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 658.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 905.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 551.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 991.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 400.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 944.T + 9.12e5T^{2} \) |
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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
−9.517334260817649104880378578712, −8.752310489528815080684260121460, −8.167841234804137116314727599163, −6.76938224732709995568089223631, −5.75464180079559561533167106880, −5.20731076192650602571202214611, −3.98084564808841431188231330959, −3.25495023640687236945049134379, −1.74014976652423621646581763814, 0,
1.74014976652423621646581763814, 3.25495023640687236945049134379, 3.98084564808841431188231330959, 5.20731076192650602571202214611, 5.75464180079559561533167106880, 6.76938224732709995568089223631, 8.167841234804137116314727599163, 8.752310489528815080684260121460, 9.517334260817649104880378578712
