| L(s) = 1 | + 4.34·2-s − 0.621·3-s + 10.8·4-s − 5.82·5-s − 2.69·6-s + 9.30·7-s + 12.3·8-s − 26.6·9-s − 25.3·10-s + 28.3·11-s − 6.74·12-s − 73.5·13-s + 40.3·14-s + 3.62·15-s − 33.0·16-s − 31.6·17-s − 115.·18-s + 49.8·19-s − 63.2·20-s − 5.78·21-s + 123.·22-s − 23·23-s − 7.69·24-s − 91.0·25-s − 319.·26-s + 33.3·27-s + 100.·28-s + ⋯ |
| L(s) = 1 | + 1.53·2-s − 0.119·3-s + 1.35·4-s − 0.521·5-s − 0.183·6-s + 0.502·7-s + 0.547·8-s − 0.985·9-s − 0.800·10-s + 0.777·11-s − 0.162·12-s − 1.56·13-s + 0.770·14-s + 0.0623·15-s − 0.516·16-s − 0.452·17-s − 1.51·18-s + 0.602·19-s − 0.707·20-s − 0.0600·21-s + 1.19·22-s − 0.208·23-s − 0.0654·24-s − 0.728·25-s − 2.40·26-s + 0.237·27-s + 0.681·28-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 | \( 1 - 4.34T + 8T^{2} \) |
| 3 | \( 1 + 0.621T + 27T^{2} \) |
| 5 | \( 1 + 5.82T + 125T^{2} \) |
| 7 | \( 1 - 9.30T + 343T^{2} \) |
| 11 | \( 1 - 28.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 73.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 31.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 49.8T + 6.85e3T^{2} \) |
| 31 | \( 1 + 222.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 65.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 4.14T + 6.89e4T^{2} \) |
| 43 | \( 1 - 185.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 397.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 313.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 181.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 688.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 241.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 343.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 367.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 921.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 324.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 726.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 891.T + 9.12e5T^{2} \) |
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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
−9.736627769040784241539351909260, −8.763711719874416657185835544145, −7.64409809190338995756783115363, −6.81617179546323017101878634613, −5.74551031820212362227115297554, −5.02502353296515105933019487297, −4.16938168766747609053886141745, −3.19321012422204939721566888952, −2.09878633084068960977705046218, 0,
2.09878633084068960977705046218, 3.19321012422204939721566888952, 4.16938168766747609053886141745, 5.02502353296515105933019487297, 5.74551031820212362227115297554, 6.81617179546323017101878634613, 7.64409809190338995756783115363, 8.763711719874416657185835544145, 9.736627769040784241539351909260
