| L(s) = 1 | + 2-s + 3-s + 4-s + 0.558·5-s + 6-s − 3.85·7-s + 8-s + 9-s + 0.558·10-s − 2.45·11-s + 12-s + 1.76·13-s − 3.85·14-s + 0.558·15-s + 16-s − 17-s + 18-s − 3.31·19-s + 0.558·20-s − 3.85·21-s − 2.45·22-s + 8.06·23-s + 24-s − 4.68·25-s + 1.76·26-s + 27-s − 3.85·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.249·5-s + 0.408·6-s − 1.45·7-s + 0.353·8-s + 0.333·9-s + 0.176·10-s − 0.738·11-s + 0.288·12-s + 0.490·13-s − 1.03·14-s + 0.144·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s − 0.760·19-s + 0.124·20-s − 0.842·21-s − 0.522·22-s + 1.68·23-s + 0.204·24-s − 0.937·25-s + 0.346·26-s + 0.192·27-s − 0.729·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 5 | \( 1 - 0.558T + 5T^{2} \) |
| 7 | \( 1 + 3.85T + 7T^{2} \) |
| 11 | \( 1 + 2.45T + 11T^{2} \) |
| 13 | \( 1 - 1.76T + 13T^{2} \) |
| 19 | \( 1 + 3.31T + 19T^{2} \) |
| 23 | \( 1 - 8.06T + 23T^{2} \) |
| 29 | \( 1 + 2.43T + 29T^{2} \) |
| 31 | \( 1 + 4.81T + 31T^{2} \) |
| 37 | \( 1 + 4.21T + 37T^{2} \) |
| 41 | \( 1 + 3.15T + 41T^{2} \) |
| 43 | \( 1 + 9.71T + 43T^{2} \) |
| 47 | \( 1 - 9.55T + 47T^{2} \) |
| 53 | \( 1 + 4.43T + 53T^{2} \) |
| 61 | \( 1 + 7.75T + 61T^{2} \) |
| 67 | \( 1 - 5.49T + 67T^{2} \) |
| 71 | \( 1 + 16.2T + 71T^{2} \) |
| 73 | \( 1 + 6.49T + 73T^{2} \) |
| 79 | \( 1 + 7.02T + 79T^{2} \) |
| 83 | \( 1 - 17.9T + 83T^{2} \) |
| 89 | \( 1 + 8.74T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−7.50224586921926541318124431456, −6.95907607947636360582341613532, −6.27576710004383164304893684528, −5.62829103186171419983891420443, −4.80622787955356808582179093350, −3.83696052561000720474770121819, −3.25829110091485406179416293394, −2.61104968770698869773037125721, −1.64189032353907375250147281203, 0,
1.64189032353907375250147281203, 2.61104968770698869773037125721, 3.25829110091485406179416293394, 3.83696052561000720474770121819, 4.80622787955356808582179093350, 5.62829103186171419983891420443, 6.27576710004383164304893684528, 6.95907607947636360582341613532, 7.50224586921926541318124431456
