L(s) = 1 | + 0.732·3-s − 5-s − 2·7-s − 2.46·9-s + 1.26·11-s + 13-s − 0.732·15-s + 3.46·17-s − 4.19·19-s − 1.46·21-s − 4.73·23-s + 25-s − 4·27-s − 9.46·29-s + 0.196·31-s + 0.928·33-s + 2·35-s − 4·37-s + 0.732·39-s − 3.46·41-s − 10.1·43-s + 2.46·45-s − 6·47-s − 3·49-s + 2.53·51-s − 10.3·53-s − 1.26·55-s + ⋯ |
L(s) = 1 | + 0.422·3-s − 0.447·5-s − 0.755·7-s − 0.821·9-s + 0.382·11-s + 0.277·13-s − 0.189·15-s + 0.840·17-s − 0.962·19-s − 0.319·21-s − 0.986·23-s + 0.200·25-s − 0.769·27-s − 1.75·29-s + 0.0352·31-s + 0.161·33-s + 0.338·35-s − 0.657·37-s + 0.117·39-s − 0.541·41-s − 1.55·43-s + 0.367·45-s − 0.875·47-s − 0.428·49-s + 0.355·51-s − 1.42·53-s − 0.170·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 - 0.732T + 3T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 1.26T + 11T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + 4.19T + 19T^{2} \) |
| 23 | \( 1 + 4.73T + 23T^{2} \) |
| 29 | \( 1 + 9.46T + 29T^{2} \) |
| 31 | \( 1 - 0.196T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 15.1T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 + 1.26T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 - 0.928T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.532663191880723444619536247441, −8.540717878316464630724059987299, −8.085339355611054246272212776661, −6.96712216720779269762232073081, −6.15940545510306890101837070572, −5.24556846747375416793353123411, −3.81607625168915196341785738898, −3.33321581223917908753820993425, −1.98037084213370900213557926370, 0,
1.98037084213370900213557926370, 3.33321581223917908753820993425, 3.81607625168915196341785738898, 5.24556846747375416793353123411, 6.15940545510306890101837070572, 6.96712216720779269762232073081, 8.085339355611054246272212776661, 8.540717878316464630724059987299, 9.532663191880723444619536247441