| L(s) = 1 | + 2.86·2-s + 3.43·3-s + 0.228·4-s + 17.9·5-s + 9.85·6-s − 32.7·7-s − 22.2·8-s − 15.1·9-s + 51.5·10-s − 26.7·11-s + 0.783·12-s − 14.4·13-s − 93.8·14-s + 61.7·15-s − 65.7·16-s − 24.7·17-s − 43.5·18-s − 94.6·19-s + 4.09·20-s − 112.·21-s − 76.6·22-s − 76.6·24-s + 197.·25-s − 41.5·26-s − 145.·27-s − 7.46·28-s − 57.5·29-s + ⋯ |
| L(s) = 1 | + 1.01·2-s + 0.661·3-s + 0.0285·4-s + 1.60·5-s + 0.670·6-s − 1.76·7-s − 0.985·8-s − 0.562·9-s + 1.63·10-s − 0.731·11-s + 0.0188·12-s − 0.309·13-s − 1.79·14-s + 1.06·15-s − 1.02·16-s − 0.352·17-s − 0.570·18-s − 1.14·19-s + 0.0458·20-s − 1.16·21-s − 0.742·22-s − 0.651·24-s + 1.58·25-s − 0.313·26-s − 1.03·27-s − 0.0503·28-s − 0.368·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 2 | \( 1 - 2.86T + 8T^{2} \) |
| 3 | \( 1 - 3.43T + 27T^{2} \) |
| 5 | \( 1 - 17.9T + 125T^{2} \) |
| 7 | \( 1 + 32.7T + 343T^{2} \) |
| 11 | \( 1 + 26.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 14.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 24.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 94.6T + 6.85e3T^{2} \) |
| 29 | \( 1 + 57.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 88.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 305.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 179.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 96.5T + 7.95e4T^{2} \) |
| 47 | \( 1 - 218.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 519.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 37.2T + 2.05e5T^{2} \) |
| 61 | \( 1 + 96.3T + 2.26e5T^{2} \) |
| 67 | \( 1 + 497.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 19.6T + 3.57e5T^{2} \) |
| 73 | \( 1 + 208.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 446.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 501.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.10e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.81e3T + 9.12e5T^{2} \) |
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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
−9.762581998660116932248667037543, −9.310737933442245987263839830678, −8.481916709005850729689331685926, −6.81626622599835596318067534148, −6.01937909236397689597919845042, −5.51132941293934779368041939068, −4.11795871920583298214171497312, −2.84490151438764186003636472226, −2.47782139645876126633744366480, 0,
2.47782139645876126633744366480, 2.84490151438764186003636472226, 4.11795871920583298214171497312, 5.51132941293934779368041939068, 6.01937909236397689597919845042, 6.81626622599835596318067534148, 8.481916709005850729689331685926, 9.310737933442245987263839830678, 9.762581998660116932248667037543
