| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 19.8·5-s − 6·6-s + 7·7-s + 8·8-s + 9·9-s − 39.6·10-s + 26.8·11-s − 12·12-s − 0.965·13-s + 14·14-s + 59.5·15-s + 16·16-s − 28.5·17-s + 18·18-s + 47.6·19-s − 79.3·20-s − 21·21-s + 53.6·22-s + 23·23-s − 24·24-s + 268.·25-s − 1.93·26-s − 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.77·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 1.25·10-s + 0.735·11-s − 0.288·12-s − 0.0206·13-s + 0.267·14-s + 1.02·15-s + 0.250·16-s − 0.407·17-s + 0.235·18-s + 0.574·19-s − 0.887·20-s − 0.218·21-s + 0.519·22-s + 0.208·23-s − 0.204·24-s + 2.14·25-s − 0.0145·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 - 7T \) |
| 23 | \( 1 - 23T \) |
| good | 5 | \( 1 + 19.8T + 125T^{2} \) |
| 11 | \( 1 - 26.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 0.965T + 2.19e3T^{2} \) |
| 17 | \( 1 + 28.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 47.6T + 6.85e3T^{2} \) |
| 29 | \( 1 - 45.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 72.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 279.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 277.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 250.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 235.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 317.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 22.3T + 2.05e5T^{2} \) |
| 61 | \( 1 + 371.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 202.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 799.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 679.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 6.87T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.15e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.04e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.49e3T + 9.12e5T^{2} \) |
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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
−9.082566212357678559247377348716, −8.202987610942197658480279170514, −7.30142691747741544327596697358, −6.79873804633380480299494634297, −5.57969572718870890017316095980, −4.60858104747284084925790783071, −4.01501907918976837401153472049, −3.08939885259549052684029271390, −1.35102953373703019929241097647, 0,
1.35102953373703019929241097647, 3.08939885259549052684029271390, 4.01501907918976837401153472049, 4.60858104747284084925790783071, 5.57969572718870890017316095980, 6.79873804633380480299494634297, 7.30142691747741544327596697358, 8.202987610942197658480279170514, 9.082566212357678559247377348716
