| L(s) = 1 | − 0.499·3-s + 10.3·5-s + 3.85·7-s − 26.7·9-s + 24.2·11-s − 13·13-s − 5.14·15-s − 56.0·17-s + 18.0·19-s − 1.92·21-s − 41.5·23-s − 18.7·25-s + 26.8·27-s + 247.·29-s + 151.·31-s − 12.0·33-s + 39.7·35-s − 205.·37-s + 6.48·39-s + 16.8·41-s − 436.·43-s − 275.·45-s − 615.·47-s − 328.·49-s + 27.9·51-s + 611.·53-s + 249.·55-s + ⋯ |
| L(s) = 1 | − 0.0960·3-s + 0.922·5-s + 0.207·7-s − 0.990·9-s + 0.664·11-s − 0.277·13-s − 0.0885·15-s − 0.800·17-s + 0.217·19-s − 0.0199·21-s − 0.377·23-s − 0.149·25-s + 0.191·27-s + 1.58·29-s + 0.879·31-s − 0.0638·33-s + 0.191·35-s − 0.912·37-s + 0.0266·39-s + 0.0640·41-s − 1.54·43-s − 0.913·45-s − 1.91·47-s − 0.956·49-s + 0.0768·51-s + 1.58·53-s + 0.612·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 3 | \( 1 + 0.499T + 27T^{2} \) |
| 5 | \( 1 - 10.3T + 125T^{2} \) |
| 7 | \( 1 - 3.85T + 343T^{2} \) |
| 11 | \( 1 - 24.2T + 1.33e3T^{2} \) |
| 17 | \( 1 + 56.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 18.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 41.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 247.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 151.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 205.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 16.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 436.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 615.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 611.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 860.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 65.6T + 2.26e5T^{2} \) |
| 67 | \( 1 - 103.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 794.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 796.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 431.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 423.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 35.9T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.18e3T + 9.12e5T^{2} \) |
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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
−8.609266686979604054809228117473, −8.019101369195856900821777981388, −6.63723469790413841506207873327, −6.34905623704395197774822146239, −5.31137001766789731183418259580, −4.62401637707923522751835672207, −3.34376756333937192784511279553, −2.38985552777768930661590435113, −1.42113946237996319593556091854, 0,
1.42113946237996319593556091854, 2.38985552777768930661590435113, 3.34376756333937192784511279553, 4.62401637707923522751835672207, 5.31137001766789731183418259580, 6.34905623704395197774822146239, 6.63723469790413841506207873327, 8.019101369195856900821777981388, 8.609266686979604054809228117473
