| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 4.31·7-s + 8-s + 9-s + 10-s + 2.05·11-s − 12-s + 1.38·13-s − 4.31·14-s − 15-s + 16-s + 18-s + 4.43·19-s + 20-s + 4.31·21-s + 2.05·22-s − 8.42·23-s − 24-s + 25-s + 1.38·26-s − 27-s − 4.31·28-s − 10.3·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 1.63·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.619·11-s − 0.288·12-s + 0.382·13-s − 1.15·14-s − 0.258·15-s + 0.250·16-s + 0.235·18-s + 1.01·19-s + 0.223·20-s + 0.942·21-s + 0.437·22-s − 1.75·23-s − 0.204·24-s + 0.200·25-s + 0.270·26-s − 0.192·27-s − 0.816·28-s − 1.92·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + 4.31T + 7T^{2} \) |
| 11 | \( 1 - 2.05T + 11T^{2} \) |
| 13 | \( 1 - 1.38T + 13T^{2} \) |
| 19 | \( 1 - 4.43T + 19T^{2} \) |
| 23 | \( 1 + 8.42T + 23T^{2} \) |
| 29 | \( 1 + 10.3T + 29T^{2} \) |
| 31 | \( 1 - 4.07T + 31T^{2} \) |
| 37 | \( 1 + 2.61T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 - 4.07T + 43T^{2} \) |
| 47 | \( 1 + 9.16T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 - 2.39T + 59T^{2} \) |
| 61 | \( 1 - 9.89T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 - 0.351T + 71T^{2} \) |
| 73 | \( 1 + 7.59T + 73T^{2} \) |
| 79 | \( 1 + 2.88T + 79T^{2} \) |
| 83 | \( 1 + 3.52T + 83T^{2} \) |
| 89 | \( 1 - 6.14T + 89T^{2} \) |
| 97 | \( 1 - 8.12T + 97T^{2} \) |
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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
−7.18877028261157930128520274960, −6.45971993368304380025333645969, −5.96452835398013382803237338347, −5.67139164960489700920029371444, −4.58914564661974515843971374444, −3.78586813507018226561995963060, −3.31617663974717064354554377660, −2.32008232393072849185635863291, −1.30490623611302837948743938066, 0,
1.30490623611302837948743938066, 2.32008232393072849185635863291, 3.31617663974717064354554377660, 3.78586813507018226561995963060, 4.58914564661974515843971374444, 5.67139164960489700920029371444, 5.96452835398013382803237338347, 6.45971993368304380025333645969, 7.18877028261157930128520274960
