L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 4.96·7-s − 8-s + 9-s − 10-s − 11-s + 12-s + 2.96·13-s + 4.96·14-s + 15-s + 16-s − 17-s − 18-s + 3.35·19-s + 20-s − 4.96·21-s + 22-s + 3.35·23-s − 24-s + 25-s − 2.96·26-s + 27-s − 4.96·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 1.87·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s + 0.821·13-s + 1.32·14-s + 0.258·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s + 0.768·19-s + 0.223·20-s − 1.08·21-s + 0.213·22-s + 0.698·23-s − 0.204·24-s + 0.200·25-s − 0.580·26-s + 0.192·27-s − 0.937·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 + 4.96T + 7T^{2} \) |
| 13 | \( 1 - 2.96T + 13T^{2} \) |
| 19 | \( 1 - 3.35T + 19T^{2} \) |
| 23 | \( 1 - 3.35T + 23T^{2} \) |
| 29 | \( 1 - 1.22T + 29T^{2} \) |
| 31 | \( 1 + 10.7T + 31T^{2} \) |
| 37 | \( 1 - 3.92T + 37T^{2} \) |
| 41 | \( 1 + 7.61T + 41T^{2} \) |
| 43 | \( 1 + 5.73T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 + 1.35T + 53T^{2} \) |
| 59 | \( 1 + 14.8T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 5.08T + 67T^{2} \) |
| 71 | \( 1 - 5.73T + 71T^{2} \) |
| 73 | \( 1 - 7.73T + 73T^{2} \) |
| 79 | \( 1 - 6.05T + 79T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 - 3.92T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−7.72832874551476421381582102022, −7.17279032333976020307162059732, −6.41355499395218221979189702013, −5.96433284287506328228688983756, −4.96654154113639069533511755869, −3.57516570331128827345029416031, −3.26987259707531233577832143560, −2.38410990198215221996065030861, −1.29540655712425178955432848904, 0,
1.29540655712425178955432848904, 2.38410990198215221996065030861, 3.26987259707531233577832143560, 3.57516570331128827345029416031, 4.96654154113639069533511755869, 5.96433284287506328228688983756, 6.41355499395218221979189702013, 7.17279032333976020307162059732, 7.72832874551476421381582102022