| L(s) = 1 | − 2-s + 2.02·3-s + 4-s + 1.22·5-s − 2.02·6-s + 1.91·7-s − 8-s + 1.09·9-s − 1.22·10-s − 3.55·11-s + 2.02·12-s − 3.23·13-s − 1.91·14-s + 2.47·15-s + 16-s − 3.39·17-s − 1.09·18-s − 1.71·19-s + 1.22·20-s + 3.88·21-s + 3.55·22-s − 5.68·23-s − 2.02·24-s − 3.50·25-s + 3.23·26-s − 3.85·27-s + 1.91·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.16·3-s + 0.5·4-s + 0.547·5-s − 0.826·6-s + 0.724·7-s − 0.353·8-s + 0.365·9-s − 0.387·10-s − 1.07·11-s + 0.584·12-s − 0.897·13-s − 0.512·14-s + 0.639·15-s + 0.250·16-s − 0.822·17-s − 0.258·18-s − 0.392·19-s + 0.273·20-s + 0.846·21-s + 0.758·22-s − 1.18·23-s − 0.413·24-s − 0.700·25-s + 0.634·26-s − 0.741·27-s + 0.362·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 - 2.02T + 3T^{2} \) |
| 5 | \( 1 - 1.22T + 5T^{2} \) |
| 7 | \( 1 - 1.91T + 7T^{2} \) |
| 11 | \( 1 + 3.55T + 11T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 + 3.39T + 17T^{2} \) |
| 19 | \( 1 + 1.71T + 19T^{2} \) |
| 23 | \( 1 + 5.68T + 23T^{2} \) |
| 29 | \( 1 - 7.78T + 29T^{2} \) |
| 31 | \( 1 + 4.38T + 31T^{2} \) |
| 37 | \( 1 - 7.53T + 37T^{2} \) |
| 41 | \( 1 + 5.40T + 41T^{2} \) |
| 43 | \( 1 + 9.17T + 43T^{2} \) |
| 47 | \( 1 - 9.42T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 - 9.07T + 59T^{2} \) |
| 61 | \( 1 + 4.13T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 + 0.472T + 73T^{2} \) |
| 79 | \( 1 - 6.86T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 + 1.70T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−8.117169474781644316874532962067, −7.76307102114811712376044295652, −6.84720476717311006085196400105, −5.94628357873751159512742102663, −5.06883317483434812944555808437, −4.21163661332702131125754270897, −3.00515076283180486270524370524, −2.29555271556035774223456268710, −1.80185316199083650374305462301, 0,
1.80185316199083650374305462301, 2.29555271556035774223456268710, 3.00515076283180486270524370524, 4.21163661332702131125754270897, 5.06883317483434812944555808437, 5.94628357873751159512742102663, 6.84720476717311006085196400105, 7.76307102114811712376044295652, 8.117169474781644316874532962067
