| L(s) = 1 | + 2.67·2-s − 1.01·3-s − 0.829·4-s + 17.5·5-s − 2.72·6-s + 31.3·7-s − 23.6·8-s − 25.9·9-s + 46.8·10-s − 68.4·11-s + 0.843·12-s − 41.0·13-s + 84.0·14-s − 17.7·15-s − 56.6·16-s − 130.·17-s − 69.5·18-s − 67.0·19-s − 14.5·20-s − 31.8·21-s − 183.·22-s − 23·23-s + 24.0·24-s + 181.·25-s − 109.·26-s + 53.8·27-s − 26.0·28-s + ⋯ |
| L(s) = 1 | + 0.946·2-s − 0.195·3-s − 0.103·4-s + 1.56·5-s − 0.185·6-s + 1.69·7-s − 1.04·8-s − 0.961·9-s + 1.48·10-s − 1.87·11-s + 0.0202·12-s − 0.875·13-s + 1.60·14-s − 0.306·15-s − 0.885·16-s − 1.86·17-s − 0.910·18-s − 0.810·19-s − 0.162·20-s − 0.331·21-s − 1.77·22-s − 0.208·23-s + 0.204·24-s + 1.45·25-s − 0.828·26-s + 0.383·27-s − 0.175·28-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 | \( 1 - 2.67T + 8T^{2} \) |
| 3 | \( 1 + 1.01T + 27T^{2} \) |
| 5 | \( 1 - 17.5T + 125T^{2} \) |
| 7 | \( 1 - 31.3T + 343T^{2} \) |
| 11 | \( 1 + 68.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 41.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 130.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 67.0T + 6.85e3T^{2} \) |
| 31 | \( 1 - 121.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 282.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 311.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 20.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + 448.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 206.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 602.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 847.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 779.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 470.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 243.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 155.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.32e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 429.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 907.T + 9.12e5T^{2} \) |
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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
−9.771615332894963395933639404113, −8.687901370445742025591861186398, −8.147958424553214424811638149148, −6.63725837178131201369798904165, −5.65137827155971186688959667966, −5.07807281193408619522314007056, −4.60183876538938390755958774392, −2.58158646274654123564257034035, −2.16276642654297359743512463199, 0,
2.16276642654297359743512463199, 2.58158646274654123564257034035, 4.60183876538938390755958774392, 5.07807281193408619522314007056, 5.65137827155971186688959667966, 6.63725837178131201369798904165, 8.147958424553214424811638149148, 8.687901370445742025591861186398, 9.771615332894963395933639404113
