| L(s) = 1 | + 2-s − 2·3-s + 4-s − 0.618·5-s − 2·6-s + 1.85·7-s + 8-s + 9-s − 0.618·10-s − 2·12-s − 6.47·13-s + 1.85·14-s + 1.23·15-s + 16-s + 5.61·17-s + 18-s + 19-s − 0.618·20-s − 3.70·21-s − 1.14·23-s − 2·24-s − 4.61·25-s − 6.47·26-s + 4·27-s + 1.85·28-s + 1.52·29-s + 1.23·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 0.5·4-s − 0.276·5-s − 0.816·6-s + 0.700·7-s + 0.353·8-s + 0.333·9-s − 0.195·10-s − 0.577·12-s − 1.79·13-s + 0.495·14-s + 0.319·15-s + 0.250·16-s + 1.36·17-s + 0.235·18-s + 0.229·19-s − 0.138·20-s − 0.809·21-s − 0.238·23-s − 0.408·24-s − 0.923·25-s − 1.26·26-s + 0.769·27-s + 0.350·28-s + 0.283·29-s + 0.225·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2T + 3T^{2} \) |
| 5 | \( 1 + 0.618T + 5T^{2} \) |
| 7 | \( 1 - 1.85T + 7T^{2} \) |
| 13 | \( 1 + 6.47T + 13T^{2} \) |
| 17 | \( 1 - 5.61T + 17T^{2} \) |
| 23 | \( 1 + 1.14T + 23T^{2} \) |
| 29 | \( 1 - 1.52T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 9.23T + 41T^{2} \) |
| 43 | \( 1 - 2.09T + 43T^{2} \) |
| 47 | \( 1 - 3.61T + 47T^{2} \) |
| 53 | \( 1 + 9.23T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 - 7.23T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 + 9.70T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 - 6.76T + 89T^{2} \) |
| 97 | \( 1 + 4.76T + 97T^{2} \) |
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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
−7.65352371865183146561316864072, −7.25990791767006919086354467654, −6.23980455174058285061701918720, −5.66772664389623827503423585486, −4.90592583676155938046473541599, −4.62349981674817245075266384114, −3.44853846450588363611909517548, −2.51323547474406339157027294835, −1.35322078436533336793995264758, 0,
1.35322078436533336793995264758, 2.51323547474406339157027294835, 3.44853846450588363611909517548, 4.62349981674817245075266384114, 4.90592583676155938046473541599, 5.66772664389623827503423585486, 6.23980455174058285061701918720, 7.25990791767006919086354467654, 7.65352371865183146561316864072
