| L(s) = 1 | + 2.14·2-s + 3-s + 2.60·4-s − 4.24·5-s + 2.14·6-s + 0.408·7-s + 1.30·8-s + 9-s − 9.11·10-s − 11-s + 2.60·12-s − 1.49·13-s + 0.877·14-s − 4.24·15-s − 2.41·16-s − 7.27·17-s + 2.14·18-s − 1.94·19-s − 11.0·20-s + 0.408·21-s − 2.14·22-s + 0.493·23-s + 1.30·24-s + 13.0·25-s − 3.21·26-s + 27-s + 1.06·28-s + ⋯ |
| L(s) = 1 | + 1.51·2-s + 0.577·3-s + 1.30·4-s − 1.89·5-s + 0.876·6-s + 0.154·7-s + 0.461·8-s + 0.333·9-s − 2.88·10-s − 0.301·11-s + 0.752·12-s − 0.415·13-s + 0.234·14-s − 1.09·15-s − 0.603·16-s − 1.76·17-s + 0.505·18-s − 0.447·19-s − 2.47·20-s + 0.0892·21-s − 0.457·22-s + 0.102·23-s + 0.266·24-s + 2.60·25-s − 0.631·26-s + 0.192·27-s + 0.201·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 2.14T + 2T^{2} \) |
| 5 | \( 1 + 4.24T + 5T^{2} \) |
| 7 | \( 1 - 0.408T + 7T^{2} \) |
| 13 | \( 1 + 1.49T + 13T^{2} \) |
| 17 | \( 1 + 7.27T + 17T^{2} \) |
| 19 | \( 1 + 1.94T + 19T^{2} \) |
| 23 | \( 1 - 0.493T + 23T^{2} \) |
| 29 | \( 1 + 2.03T + 29T^{2} \) |
| 31 | \( 1 + 0.979T + 31T^{2} \) |
| 37 | \( 1 + 2.19T + 37T^{2} \) |
| 41 | \( 1 + 3.81T + 41T^{2} \) |
| 43 | \( 1 - 1.15T + 43T^{2} \) |
| 47 | \( 1 - 6.83T + 47T^{2} \) |
| 53 | \( 1 + 4.05T + 53T^{2} \) |
| 59 | \( 1 + 5.72T + 59T^{2} \) |
| 67 | \( 1 + 8.88T + 67T^{2} \) |
| 71 | \( 1 - 7.77T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 - 7.10T + 83T^{2} \) |
| 89 | \( 1 + 8.88T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−8.568762308592098346737299589921, −7.85697204157360787194443489134, −7.06786031104710795306043549283, −6.48474965779198940231529743174, −5.09524450245318091664541319029, −4.49591757611816152597373365887, −3.91392947082222031262999472626, −3.15876561303566728157680631071, −2.22665362466799001067130924656, 0,
2.22665362466799001067130924656, 3.15876561303566728157680631071, 3.91392947082222031262999472626, 4.49591757611816152597373365887, 5.09524450245318091664541319029, 6.48474965779198940231529743174, 7.06786031104710795306043549283, 7.85697204157360787194443489134, 8.568762308592098346737299589921
