| L(s) = 1 | + 2·2-s + 0.130·3-s + 4·4-s + 0.260·6-s + 5.61·7-s + 8·8-s − 26.9·9-s − 7.81·11-s + 0.521·12-s − 35.9·13-s + 11.2·14-s + 16·16-s + 17·17-s − 53.9·18-s + 103.·19-s + 0.732·21-s − 15.6·22-s − 167.·23-s + 1.04·24-s − 71.9·26-s − 7.04·27-s + 22.4·28-s − 138.·29-s − 103.·31-s + 32·32-s − 1.01·33-s + 34·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.0250·3-s + 0.5·4-s + 0.0177·6-s + 0.303·7-s + 0.353·8-s − 0.999·9-s − 0.214·11-s + 0.0125·12-s − 0.768·13-s + 0.214·14-s + 0.250·16-s + 0.242·17-s − 0.706·18-s + 1.25·19-s + 0.00761·21-s − 0.151·22-s − 1.52·23-s + 0.00887·24-s − 0.543·26-s − 0.0501·27-s + 0.151·28-s − 0.884·29-s − 0.600·31-s + 0.176·32-s − 0.00537·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 2T \) |
| 5 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 3 | \( 1 - 0.130T + 27T^{2} \) |
| 7 | \( 1 - 5.61T + 343T^{2} \) |
| 11 | \( 1 + 7.81T + 1.33e3T^{2} \) |
| 13 | \( 1 + 35.9T + 2.19e3T^{2} \) |
| 19 | \( 1 - 103.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 167.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 138.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 103.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 45.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 313.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 147.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 486.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 348.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 477.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 865.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 473.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 69.8T + 3.57e5T^{2} \) |
| 73 | \( 1 - 155.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 917.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 232.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 970.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.17e3T + 9.12e5T^{2} \) |
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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
−9.528127754475387269178070601328, −8.310512575323517425972992108202, −7.68355657644644267502160402683, −6.67627061225128694409837470384, −5.57072884814288021599203963323, −5.11935895515225913326928244924, −3.82899178576225730374153843146, −2.89898130382695399192662767974, −1.78603561708107219488295041736, 0,
1.78603561708107219488295041736, 2.89898130382695399192662767974, 3.82899178576225730374153843146, 5.11935895515225913326928244924, 5.57072884814288021599203963323, 6.67627061225128694409837470384, 7.68355657644644267502160402683, 8.310512575323517425972992108202, 9.528127754475387269178070601328
