L(s) = 1 | + 2-s − 2.58·3-s + 4-s − 5-s − 2.58·6-s − 0.667·7-s + 8-s + 3.67·9-s − 10-s − 2.84·11-s − 2.58·12-s + 13-s − 0.667·14-s + 2.58·15-s + 16-s + 1.37·17-s + 3.67·18-s + 1.12·19-s − 20-s + 1.72·21-s − 2.84·22-s − 0.435·23-s − 2.58·24-s + 25-s + 26-s − 1.74·27-s − 0.667·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.49·3-s + 0.5·4-s − 0.447·5-s − 1.05·6-s − 0.252·7-s + 0.353·8-s + 1.22·9-s − 0.316·10-s − 0.858·11-s − 0.745·12-s + 0.277·13-s − 0.178·14-s + 0.667·15-s + 0.250·16-s + 0.333·17-s + 0.866·18-s + 0.258·19-s − 0.223·20-s + 0.376·21-s − 0.606·22-s − 0.0908·23-s − 0.527·24-s + 0.200·25-s + 0.196·26-s − 0.335·27-s − 0.126·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
good | 3 | \( 1 + 2.58T + 3T^{2} \) |
| 7 | \( 1 + 0.667T + 7T^{2} \) |
| 11 | \( 1 + 2.84T + 11T^{2} \) |
| 17 | \( 1 - 1.37T + 17T^{2} \) |
| 19 | \( 1 - 1.12T + 19T^{2} \) |
| 23 | \( 1 + 0.435T + 23T^{2} \) |
| 29 | \( 1 - 3.39T + 29T^{2} \) |
| 37 | \( 1 - 6.66T + 37T^{2} \) |
| 41 | \( 1 + 4.55T + 41T^{2} \) |
| 43 | \( 1 + 8.44T + 43T^{2} \) |
| 47 | \( 1 - 1.87T + 47T^{2} \) |
| 53 | \( 1 - 6.09T + 53T^{2} \) |
| 59 | \( 1 - 9.16T + 59T^{2} \) |
| 61 | \( 1 - 0.621T + 61T^{2} \) |
| 67 | \( 1 + 15.1T + 67T^{2} \) |
| 71 | \( 1 + 0.573T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 + 8.71T + 79T^{2} \) |
| 83 | \( 1 + 14.9T + 83T^{2} \) |
| 89 | \( 1 - 0.261T + 89T^{2} \) |
| 97 | \( 1 + 14.0T + 97T^{2} \) |
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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
−7.85613470943149432610925366501, −7.07568853474646279051724983248, −6.43908347625555048483282982968, −5.73820840724487134521860758098, −5.14607117783374098453494832861, −4.51473091711273147743596962466, −3.58381008013633195530218147078, −2.64860878177217478320781577730, −1.22298998758999407745186170780, 0,
1.22298998758999407745186170780, 2.64860878177217478320781577730, 3.58381008013633195530218147078, 4.51473091711273147743596962466, 5.14607117783374098453494832861, 5.73820840724487134521860758098, 6.43908347625555048483282982968, 7.07568853474646279051724983248, 7.85613470943149432610925366501