L(s) = 1 | + 2-s + 3-s + 4-s + 1.47·5-s + 6-s − 0.770·7-s + 8-s + 9-s + 1.47·10-s + 2.85·11-s + 12-s + 4.25·13-s − 0.770·14-s + 1.47·15-s + 16-s + 17-s + 18-s + 0.374·19-s + 1.47·20-s − 0.770·21-s + 2.85·22-s + 0.425·23-s + 24-s − 2.82·25-s + 4.25·26-s + 27-s − 0.770·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.659·5-s + 0.408·6-s − 0.291·7-s + 0.353·8-s + 0.333·9-s + 0.466·10-s + 0.860·11-s + 0.288·12-s + 1.18·13-s − 0.205·14-s + 0.380·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s + 0.0858·19-s + 0.329·20-s − 0.168·21-s + 0.608·22-s + 0.0888·23-s + 0.204·24-s − 0.564·25-s + 0.834·26-s + 0.192·27-s − 0.145·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.187622049\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.187622049\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
good | 5 | \( 1 - 1.47T + 5T^{2} \) |
| 7 | \( 1 + 0.770T + 7T^{2} \) |
| 11 | \( 1 - 2.85T + 11T^{2} \) |
| 13 | \( 1 - 4.25T + 13T^{2} \) |
| 19 | \( 1 - 0.374T + 19T^{2} \) |
| 23 | \( 1 - 0.425T + 23T^{2} \) |
| 29 | \( 1 + 1.19T + 29T^{2} \) |
| 31 | \( 1 + 2.96T + 31T^{2} \) |
| 37 | \( 1 - 6.00T + 37T^{2} \) |
| 41 | \( 1 - 4.04T + 41T^{2} \) |
| 43 | \( 1 - 0.816T + 43T^{2} \) |
| 47 | \( 1 + 0.227T + 47T^{2} \) |
| 53 | \( 1 + 1.29T + 53T^{2} \) |
| 61 | \( 1 - 8.62T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 - 5.49T + 71T^{2} \) |
| 73 | \( 1 - 1.56T + 73T^{2} \) |
| 79 | \( 1 - 0.251T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + 3.50T + 97T^{2} \) |
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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
−8.102570044493055011344918339001, −7.23242120417753076480276206952, −6.50246740127851645343726917733, −5.97968790719381891997063988711, −5.30120980124290771466811261960, −4.18805759403164163051916326091, −3.72960245716365426578509602460, −2.90387830257785157057806694601, −1.96852118218628641065307880619, −1.14814927712675766038910427096,
1.14814927712675766038910427096, 1.96852118218628641065307880619, 2.90387830257785157057806694601, 3.72960245716365426578509602460, 4.18805759403164163051916326091, 5.30120980124290771466811261960, 5.97968790719381891997063988711, 6.50246740127851645343726917733, 7.23242120417753076480276206952, 8.102570044493055011344918339001