| L(s) = 1 | + 2.83·2-s + 0.917·3-s + 0.0233·4-s + 12.0·5-s + 2.60·6-s + 32.1·7-s − 22.5·8-s − 26.1·9-s + 34.2·10-s − 57.7·11-s + 0.0213·12-s + 6.32·13-s + 91.0·14-s + 11.0·15-s − 64.1·16-s − 74.8·17-s − 74.0·18-s − 89.0·19-s + 0.281·20-s + 29.5·21-s − 163.·22-s − 149.·23-s − 20.7·24-s + 21.2·25-s + 17.9·26-s − 48.7·27-s + 0.749·28-s + ⋯ |
| L(s) = 1 | + 1.00·2-s + 0.176·3-s + 0.00291·4-s + 1.08·5-s + 0.176·6-s + 1.73·7-s − 0.998·8-s − 0.968·9-s + 1.08·10-s − 1.58·11-s + 0.000514·12-s + 0.134·13-s + 1.73·14-s + 0.191·15-s − 1.00·16-s − 1.06·17-s − 0.970·18-s − 1.07·19-s + 0.00315·20-s + 0.306·21-s − 1.58·22-s − 1.35·23-s − 0.176·24-s + 0.169·25-s + 0.135·26-s − 0.347·27-s + 0.00505·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 31 | \( 1 \) |
| good | 2 | \( 1 - 2.83T + 8T^{2} \) |
| 3 | \( 1 - 0.917T + 27T^{2} \) |
| 5 | \( 1 - 12.0T + 125T^{2} \) |
| 7 | \( 1 - 32.1T + 343T^{2} \) |
| 11 | \( 1 + 57.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 6.32T + 2.19e3T^{2} \) |
| 17 | \( 1 + 74.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 89.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 149.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 96.2T + 2.43e4T^{2} \) |
| 37 | \( 1 + 30.4T + 5.06e4T^{2} \) |
| 41 | \( 1 + 17.5T + 6.89e4T^{2} \) |
| 43 | \( 1 - 249.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 108.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 539.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 206.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 388.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 217.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 172.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 439.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 850.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 766.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 650.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 910.T + 9.12e5T^{2} \) |
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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
−8.999255205000790470044494056216, −8.463712879778396191180648111807, −7.66495466487444839763881788281, −6.12903881421082673957625452011, −5.59536895557765610300322596075, −4.91528833954200276655130393978, −4.07177186873180456350958827183, −2.50700638725581893214719283081, −2.06560352889465595124572739662, 0,
2.06560352889465595124572739662, 2.50700638725581893214719283081, 4.07177186873180456350958827183, 4.91528833954200276655130393978, 5.59536895557765610300322596075, 6.12903881421082673957625452011, 7.66495466487444839763881788281, 8.463712879778396191180648111807, 8.999255205000790470044494056216
