L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 3.04·7-s + 8-s + 9-s + 10-s + 1.13·11-s − 12-s + 3.04·14-s − 15-s + 16-s + 2.69·17-s + 18-s + 1.55·19-s + 20-s − 3.04·21-s + 1.13·22-s + 5.40·23-s − 24-s + 25-s − 27-s + 3.04·28-s − 6.15·29-s − 30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s + 1.15·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.342·11-s − 0.288·12-s + 0.814·14-s − 0.258·15-s + 0.250·16-s + 0.652·17-s + 0.235·18-s + 0.356·19-s + 0.223·20-s − 0.665·21-s + 0.242·22-s + 1.12·23-s − 0.204·24-s + 0.200·25-s − 0.192·27-s + 0.576·28-s − 1.14·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.666916359\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.666916359\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 - 3.04T + 7T^{2} \) |
| 11 | \( 1 - 1.13T + 11T^{2} \) |
| 17 | \( 1 - 2.69T + 17T^{2} \) |
| 19 | \( 1 - 1.55T + 19T^{2} \) |
| 23 | \( 1 - 5.40T + 23T^{2} \) |
| 29 | \( 1 + 6.15T + 29T^{2} \) |
| 31 | \( 1 + 0.246T + 31T^{2} \) |
| 37 | \( 1 + 0.554T + 37T^{2} \) |
| 41 | \( 1 - 8.26T + 41T^{2} \) |
| 43 | \( 1 - 2.03T + 43T^{2} \) |
| 47 | \( 1 + 6.70T + 47T^{2} \) |
| 53 | \( 1 - 5.77T + 53T^{2} \) |
| 59 | \( 1 - 1.36T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 + 2.47T + 67T^{2} \) |
| 71 | \( 1 + 8.57T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 79 | \( 1 - 5.33T + 79T^{2} \) |
| 83 | \( 1 + 5.45T + 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 - 1.93T + 97T^{2} \) |
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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
−8.029200954619536177995824479130, −7.37007816334278611137118320036, −6.73417003395807437439405756773, −5.75995452705764168383936045021, −5.39280135009957791750937130881, −4.67126656860582450042478400239, −3.92327484355580957450920794546, −2.90433027097196402728531525410, −1.83540998482046399607611568812, −1.05949674760602724186523704552,
1.05949674760602724186523704552, 1.83540998482046399607611568812, 2.90433027097196402728531525410, 3.92327484355580957450920794546, 4.67126656860582450042478400239, 5.39280135009957791750937130881, 5.75995452705764168383936045021, 6.73417003395807437439405756773, 7.37007816334278611137118320036, 8.029200954619536177995824479130